Frobenius Local Rings of Length 5 and Index of Nilpotency 3

Abstract

This paper investigates finite local non-chain rings associated by the well-known invariants $p,n,m,l,$ and k, where p is a prime number. In particular, we provide a comprehensive characterization of Frobenius local rings of length $l=5$ and index of nilpotency $t=3$, where t the index of nilpotency of the maximal ideal. The relevance of Frobenius rings is notable in coding theory, as it has been demonstrated that two classical results by MacWilliams—the Extension Theorem and the MacWilliams identities—are applicable not only to finite fields but also to finite Frobenius rings. We, therefore, classify and count Frobenius local rings of order $p^{5m}$ with $t=3,$ outlining their properties in connection with various values of $n.$

1. Introduction

Finite local rings represent a significant category within the realm of abstract algebra, characterized by unique properties that have garnered considerable attention from mathematicians. By definition, a finite local ring is a finite ring in which every ideal is contained within a single maximal ideal [1,2,3,4,5]. This structure markedly differs from that of more general rings, which may feature multiple maximal ideals, resulting in a more complex ideal hierarchy. A foundational example of finite local rings can be observed in Galois rings, first introduced by Krull [6]. These rings, particularly ${\mathbb{Z}}_{p^n}$, where p is a prime number, exemplify the principles underlying finite local rings. The examination of these rings is intricately linked to module theory, as finitely produced modules over finite local rings demonstrate unique characteristics that can illuminate the rings’ comprehensive structure.

Let $\mathfrak{M}$ be a module over a ring R. The socle of $\mathfrak{M}$, denoted as $\mathrm{soc}\left(\mathfrak{M}\right)$, is defined as the total sum of all minimal R-submodules of $\mathfrak{M}$. A ring R is classified as Frobenius if $\mathrm{soc}\left(R\right)\cong {\displaystyle \frac{R}{N}}={\mathbb{F}}_{p^m}$ when R is considered as an R-module. Recent discoveries indicate that Frobenius rings can function as alphabets for linear codes, owing to the significance of classical theorems formulated by MacWilliams—specifically, the MacWilliams identities and Extension Theorem. These theorems are applicable to Frobenius local rings in a manner comparable to their relevance in finite fields. For further exploration of this subject, refer to [7,8,9,10,11,12].

This article assumes that all rings considered are finite, commutative, and contain an identity. Furthermore, all modules over these rings are regarded as finitely produced. Any finite local ring R can be represented as a quotient of a polynomial ring over its primary ring $\mathcal{Q}$ in the following way:

$$R\cong {\displaystyle \frac{\mathcal{Q}[{\theta }_1,{\theta }_2,\cdots ,{\theta }_k]}{\mathfrak{I}}},$$

where $\mathfrak{I}$ is an ideal associated with ${\theta }_i$. The maximal ideal can be defined as $N={\displaystyle \frac{\langle p,{\theta }_1,{\theta }_2,\dots ,{\theta }_k\rangle }{\mathfrak{I}}}$. The residue field is expressed as ${\displaystyle \frac{R}{N}}\cong {\mathbb{F}}_{p^m}$ for some integer m. The magnitude of R is defined as $|{\mathbb{F}}_{p^m}|={\left(p^m\right)}^l$, where l denotes the length of R as an R-module. Given that R is finite, there exists a positive integer t for which $N^t=0$ and $N^{t-1}\ne 0$; this t is referred to as the index of nilpotency of N, with the condition that $t\le l$. In addition, it is known that ${\mathrm{o}}_{+}\left(1\right)$ (additive order) is represented as $p^n$ for a certain n, referred to as the characteristic of R. The classification of Frobenius local rings essentially entails examining the structure of the ideal I, resulting in the determination that these rings may generally be expressed as a direct sum of their prime components subrings $\mathcal{Q}=GR(p^n,m)$ of the form

$$R=\mathcal{Q}+{\theta }_1\mathcal{Q}+\cdots +{\theta }_k\mathcal{Q}$$

for a certain positive integer k. The parameters p, n, m, k, and l that are associated with R are designated as the parameters of R. This study aims to find and categorize all Frobenius local rings with fixed variables p, n, m, k, and $l=5$, especially those with $t=3$.

It is commonly known that, up to isomorphism, there is a unique local ring with p elements for a prime integer p: the Galois field ${\mathbb{F}}_p$. However, local rings of order $p^2$ encompass ${\mathbb{F}}_{p^2}$, ${\mathbb{Z}}_{p^2}$, and ${\displaystyle \frac{{\mathbb{F}}_p\left[\theta \right]}{\langle {\theta }^2\rangle }}$. When p is an odd prime, the local rings containing $p^3$ elements are ${\mathbb{F}}_{p^3}$, ${\mathbb{Z}}_{p^3}$, ${\displaystyle \frac{{\mathbb{F}}_p\left[\theta \right]}{\langle {\theta }^3\rangle }}$, ${\displaystyle \frac{{\mathbb{Z}}_{p^2}\left[\theta \right]}{\langle {\theta }^2-p,p\theta \rangle }}$, ${\displaystyle \frac{{\mathbb{Z}}_{p^2}\left[\theta \right]}{\langle {\theta }^2-p\xi ,p\theta \rangle }}$ (where $\xi$ is a primitive element of ${\mathbb{F}}_p$), ${\displaystyle \frac{{\mathbb{F}}_p[\theta ,\pi ]}{{\langle \theta ,\pi \rangle }^2}}$, and ${\displaystyle \frac{{\mathbb{Z}}_{p^2}\left[\theta \right]}{\langle {\theta }^2,p\theta \rangle }}$. In the case where $p=2$, the local rings containing $2^3$ elements include ${\mathbb{F}}_{2^3}$, ${\mathbb{Z}}_{2^3}$, ${\displaystyle \frac{{\mathbb{F}}_2\left[\theta \right]}{\langle {\theta }^3,2\theta \rangle }}$, ${\displaystyle \frac{{\mathbb{Z}}_{2^2}\left[\theta \right]}{\langle {\theta }^2-2,2\theta \rangle }}$, ${\displaystyle \frac{{\mathbb{F}}_2[\theta ,\pi ]}{{\langle \theta ,\pi \rangle }^2}}$, and ${\displaystyle \frac{{\mathbb{Z}}_{2^2}\left[\theta \right]}{\langle {\theta }^2,2\theta \rangle }}$.

The rings ${\displaystyle \frac{{\mathbb{F}}_p[\theta ,\pi ]}{{\langle \theta ,\pi \rangle }^2}}$, ${\displaystyle \frac{{\mathbb{Z}}_{p^2}\left[\theta \right]}{\langle {\theta }^2,p\theta \rangle }}$, ${\displaystyle \frac{{\mathbb{F}}_2[\theta ,\pi ]}{{\langle \theta ,\pi \rangle }^2}}$, and ${\displaystyle \frac{{\mathbb{Z}}_{2^2}\left[\theta \right]}{\langle {\theta }^2,2\theta \rangle }}$ are not classified as Frobenius local rings, since ann$\left(N\right)$ is not simple, while the remaining rings qualify as Frobenius. In [13], a partial study of local rings was undertaken for the case of $m=1$. Furthermore, the Frobenius local ring with $p^4$ elements was introduced in [11], where linear codes were explored. Investigating the family of finite Frobenius local rings with $p^{5m}$ elements for $m>1$ would be a fascinating area of research. This study aims to elucidate the identification of Frobenius local rings with length 5 and index of nilpotency 3.

The organization of this study is as follows: Section 2 elucidates fundamental notions pertaining to finite local rings and their corresponding modules. Following that, in Section 3, we detail our comprehensive analysis and determination of all Frobenius local rings with $p^{5m}$ elements, where $t=3$ and p represents a prime number. This section is further subdivided into subsections based on the values of n, specifically addressing the cases where n equals 1, 2, or 3.

2. Preliminaries

In this discussion, R and ℜ represent rings, while $\mathfrak{M}$ denotes an R-module. When $R\subseteq \Re$, the structure ℜ is designated as an extension of R. The ideal $\mathfrak{I}\Re$ represents the extension of $\mathfrak{I}$ to ℜ, with $\mathfrak{I}$ being an ideal of R. Furthermore, for any R-module $\mathfrak{M}$, we define

$${\mathrm{ann}}_R\left(\mathfrak{M}\right)=\{r\in R:ra=0\mathrm{for}\mathrm{all}a\in \mathfrak{M}\}.$$

Moreover, the ${\displaystyle \frac{R}{\mathfrak{I}}}$-module structure on $\mathfrak{M}$ arises naturally for any ideal $\mathfrak{I}\subset {\mathrm{ann}}_R\left(\mathfrak{M}\right)$. The scalar multiplication is given by

$$(r+\mathfrak{I})a=ra,$$

for $r\in R$ and $a\in \mathfrak{M}$, thereby ensuring that the lattices of R-submodules and ${\displaystyle \frac{R}{\mathfrak{I}}}$-submodules of $\mathfrak{M}$ coincide. A proper inclusion sequence of R-submodules of $\mathfrak{M}$ is denoted as a series, expressed as

$$\mathfrak{M}={\mathfrak{M}}_0\supset {\mathfrak{M}}_1\supset \cdots \supset {\mathfrak{M}}_{l-1}\supset {\mathfrak{M}}_l=0.$$

The length of this chain is defined as l, which corresponds to the number of inclusions. This chain is called a composition series when each quotient ${\displaystyle \frac{{\mathfrak{M}}_i}{{\mathfrak{M}}_{i+1}}}$ does not contain any non-zero simple module. Moreover, the length, denoted $l_R\left(\mathfrak{M}\right)$, is infinite if $\mathfrak{M}$ lacks a finite composition series. It is noteworthy that the lengths of such series remain invariant under isomorphism. For an ${\mathbb{F}}_{p^m}$-module, the length corresponds to the dimension of the associated ${\mathbb{F}}_{p^m}$-vector space.

A ring R with a sole maximal ideal N is called local, and the residue field ${\displaystyle \frac{R}{N}}={\mathbb{F}}_{p^m}$. As stated by Nakayama’s Lemma ([14]), R possesses a chain of ideals, characterized by an integer $t\ge 1$, known as the nilpotency index of N, such that

$$R\supset N\supset N^2\supset \dots \supset N^{t-1}\supset N^t=0.$$

This indicates that $t\le l_R\left(R\right)$. Furthermore, the size of $\mathfrak{M}$ can be expressed as

$$\left|\mathfrak{M}\right|={p^m}^{l_R\left(\mathfrak{M}\right)}.$$

As direct consequences, $\left|R\right|=p^{lm}$ and $\left|N\right|=p^{(l-1)m}\mathrm{provided}{N}^l=0.$ In R, the characteristic takes the form $p^n$, where $1\le n\le l$. Additionally, there is a prime subring $\mathcal{Q}=GR(p^n,m)$, called a Galois ring with parameters $p,n,m$. There exist elements ${\theta }_i\in N$, where $1\le i\le k$, such that (as a $\mathcal{Q}$-module)

$$\begin{array}{cc}\hfill R& =\sum _{i=0}^k{\theta }_i\mathcal{Q},\hfill \\ \hfill N& =p\mathcal{Q}+\sum _{i=1}^k{\theta }_i\mathcal{Q}.\hfill \end{array}$$

(1)

The set of elements $\{{\theta }_1,{\theta }_2,\dots ,{\theta }_k\}$ is called a distinguished basis of R over $\mathcal{Q}$ [1]. These elements will be crucial, as will be seen later, in determining such rings, under isomorphism, with parameters $p,n,m,l$, and k. Observe that when $k=0,$ then $R=\mathcal{Q}=GR(p^n,m).$ Moreover, if $t=1,$ then $N=0,$ and thus R is a field, i.e., $l=1.$ Furthermore, if $k=1,$ then such rings are called singleton local rings and they have been fully characterized in [15]. Therefore, in our work, we have always $t>2.$ A special class of local rings arises when $n=l$. This class contains Galois rings $GR(p^n,m)$, which are uniquely determined with respect to the parameters $p,n$, and m. Moreover, this class is constructed over ${\mathbb{Z}}_{p^n}$ as follows. If $N=\langle p\rangle$, we obtain $n=l$ and

$$R=\mathcal{Q}=GR(p^n,m)\cong {\mathbb{Z}}_{p^n}\left[\xi \right]\cong {\displaystyle \frac{{\mathbb{Z}}_{p^n}\left[x\right]}{\langle g\left(x\right)\rangle }},$$

where $\xi$ is a root of a certain polynomial $g\left(x\right)\in {\mathbb{Z}}_{p^n}\left[x\right]$ of degree m.

A finite local ring R is termed Frobenius if the annihilator of the Jacobson radical ${\mathrm{ann}}_R\left(N\right)$ stands as the unique minimal ideal of R. This is equivalent to saying that ${\mathrm{ann}}_R\left(N\right)$ constitutes a simple ideal of R (see [12]). An alternative characterization of a Frobenius ring employs the concept of the socle of an R-module $\mathfrak{M}$. The socle, denoted as $\mathrm{soc}\left(\mathfrak{M}\right)$, is defined as the sum of all minimal R-submodules of $\mathfrak{M}$. In [16], a ring R is identified as Frobenius if the quotient ${\displaystyle \frac{R}{N}}$ is isomorphic to the socle of R, and this occurs if and only if $\mathrm{soc}\left(R\right)$ is cyclic. For a finite Frobenius local ring R, it follows that ${\mathrm{ann}}_R\left(N\right)=N^{t-1}$. However, it is noteworthy that the converse does not hold universally, as discussed later in this work. The collection of Frobenius rings is quite broad, encompassing both finite chain rings and non-chain rings, such as

$${\displaystyle \frac{{\mathbb{Z}}_p[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2\rangle }}\mathrm{and}{\displaystyle \frac{{\mathbb{Z}}_{p^n}\left[\theta \right]}{\langle {\theta }^2-p^{n-1},p\theta \rangle }},\mathrm{for}n\ge 3.$$

For additional examples, we refer the reader to [11].

Let us define $\Re ={\displaystyle \frac{R\left[x\right]}{\langle g\left(x\right)\rangle }}=\{r_0+r_{1}x+\cdots +r_s{x}^s:r_i\in R\}$, where $s=\mathrm{deg}\left(g\right)$. This ring ℜ serves as a separable extension of R and is also local, with the maximal ideal represented by $NT$ and having the residue field ${\mathbb{F}}_{p^{ms}}$ (see [17]). Supposing $\mathfrak{I}$ is an ideal of R, we have

$$l_R\left(\mathfrak{I}\right)=l_{\Re }(\mathfrak{I}\Re ),\mathrm{and}\left(\mathrm{ann}\left(\mathfrak{I}\right)\right)\Re ={\mathrm{ann}}_{\Re }(\mathfrak{I}\Re ).$$

From these facts, it follows that:

$$l\left(\mathrm{ann}\left(N\right)\right)=l_{\Re }\left({\mathrm{ann}}_{\Re }(N\Re )\right),$$

and

$$l\left({\displaystyle \frac{N}{N^2}}\right)=l_{\Re }\left({\displaystyle \frac{N\Re }{N^2\Re }}\right)=v_{\Re }(N\Re ).$$

Therefore, R is Frobenius when

$$l\left(\mathrm{ann}\left(N\right)\right)=l_{\Re }\left({\mathrm{ann}}_{\Re }(N\Re )\right)=1,$$

which is equivalent to ℜ being a Frobenius ring. Moreover, if we have a minimal generating set $\{{\theta }_1,\dots ,{\theta }_l\}$ for N, this set also serves as a minimal generating set for $N\Re$ over ℜ. It is important to note that, when $R={\mathbb{Z}}_{p^n}$, the ring ℜ is specifically the Galois ring $GR(p^n,s)$.

3. Frobenius Local Rings of Length 5

Let R be a Frobenius local ring characterized by parameters $p,n,m,5,$ and k. In this section, we will demonstrate all possible structures of R and their enumeration by varying the values of n and k. Let $\mathfrak{M}$ be an R-module. A subset S of $\mathfrak{M}$ spans $\mathfrak{M}$ if and only if its image $\overline{S}$ in ${\displaystyle \frac{\mathfrak{M}}{N\mathfrak{M}}}$ generates ${\displaystyle \frac{\mathfrak{M}}{N\mathfrak{M}}}$. The set of generators of $\mathfrak{M}$ is obtained by lifting a basis of the ${\displaystyle \frac{R}{N}}$-vector space. The expression ${\displaystyle \frac{\mathfrak{M}}{N\mathfrak{M}}}$ represents a minimal generating set for $\mathfrak{M}$ over the ring R, while ${\vartheta }_R\left(\mathfrak{M}\right)$ indicates the count of these generators. It is important to observe that

$${\vartheta }_R\left(\mathfrak{M}\right)={\dim }_{{\mathbb{F}}_{p^m}}\left({\displaystyle \frac{\mathfrak{M}}{N\mathfrak{M}}}\right)=l_R\left({\displaystyle \frac{\mathfrak{M}}{N\mathfrak{M}}}\right).$$

(2)

In particular, we have

$$\vartheta \left(N\right)=l\left({\displaystyle \frac{N}{N^2}}\right).$$

(3)

To demonstrate a relationship among $t,\vartheta \left(N\right)$, and k, which is essential for establishing the key conclusions of this section, the following results are critical.

Lemma 1

([9]).Suppose that R is a local ring with $p,n,m,5,k.$ Thus,

$$\lfloor {\displaystyle \frac{5}{n}}\rfloor -1\le k\le 5-n.$$

(4)

The configuration and algebraic framework of R are wholly dictated by the parameters $n,l,k.$ Subsequently, we formulate a significant relationship between the parameters n and k of a local ring, which will elucidate the construction of such a ring.

Proposition 1.

Let R be a local ring with $l=5$ and $t=3.$ Then,

$$k=\left\{\begin{array}{c}4,\mathit{if}n=1;\hfill \\ \left\{\begin{array}{cc}3,\hfill & \mathit{if}p\in N^2;\hfill \\ 2,\hfill & \mathit{if}p\in N\setminus N^2;\hfill \end{array}\right.(n=2)\hfill \\ 2,\mathit{if}n=3.\hfill \end{array}\right.$$

In particular,$\vartheta \left(N\right)=3.$

Proof. 

First observe $N^2\ne 0$ and $N^3=0.$ Moreover, it is clear that $\vartheta \left(N\right)\le k+1.$ The ideal $N^2$ is generated by at most ${\left(\vartheta \left(N\right)\right)}^2$ elements of $N,$ and so $\vartheta \left(N^2\right)\le {\left(\vartheta \left(N\right)\right)}^2.$ Thus, we have $\vartheta \left(N\right)+\vartheta \left(N^2\right)\ge k+1.$ When $n=1,$ we must have $\vartheta \left(N\right)+\vartheta \left(N^2\right)=k=4,$ and hence $\vartheta \left(N\right)=3$ and $\vartheta \left(N^2\right)=1$ or $\vartheta \left(N\right)=2$ and $\vartheta \left(N^2\right)=2$ in light of Lemma 1. So, we have the result if $n=1.$ However, again by Lemma 1, $1\le k\le 3$ in case of $n=2.$ Assume $k=1,$ so $R=\mathcal{Q}+u\mathcal{Q},$ where $\mathcal{Q}=GR(p^2,m).$ As R is of order $p^{5m},$ then there must be $5=2+t_{\theta },$ where $o_{+}\left(\theta \right)=p^{t_{\theta }}.$ Note that $t_{\theta }\le 2,$ and hence there is no such ring with $n=2$ and $k=1.$ Now, if $k=2,$ this means that

$$R=\mathcal{Q}+\theta \mathcal{Q}+\pi \mathcal{Q}.$$

The order of R explains that only $p\theta$ or $p\pi$ must be $0.$ This implies we cannot have p in ann$\left(N\right)$ when $k=2.$ As R is Frobenius, we imply that $p\in N\setminus N^2.$ But if $k=3,$ then

$$R=\mathcal{Q}+\theta \mathcal{Q}+\pi \mathcal{Q}+w\mathcal{Q}.$$

Again, the order of R shows that p must be in ann$\left(N\right).$ Finally, let $n=3;$ clearly, $k=2$ by Lemma 1. Note that $p\in N^2$ in this case. Observe that k cannot take $1,$ by a similar argument to the one above. □

Remark 1.

When $t=3$ and $N=\langle p,\theta ,\pi \rangle ,$ then θ and π hold the equations ${\theta }^3={\pi }^3=\theta {\pi }^2={\theta }^2\pi =0.$ As $t=3,$ then we have $N^3=0.$ The elements ${\theta }^3,{\pi }^3,\theta {\pi }^2$, and ${\theta }^2\pi$ are in $N^3,$ and thus they are zeros.

This section now presents the primary outcomes. Before proceeding with the proof, we observe the following. Recall that $\mid R\mid =p^{5m}$. Additionally, since the nilpotency index of N is $t=3$, it follows that char$\left(R\right)\in \{p,p^2,p^3\}$. Therefore, we divide the section into subsections according to $n.$

(i)

If $n=1,$ then $\mathcal{Q}={\mathbb{F}}_{p^m}.$ In this case,

$$R=\mathcal{Q}+\theta \mathcal{Q}+\pi \mathcal{Q}+w\mathcal{Q}+\zeta \mathcal{Q}.$$

(5)

(ii)

If $n=2,$ then we have $\mathcal{Q}=GR(p^2,m).$ By Proposition 1,

$$R=\left\{\begin{array}{cc}\mathcal{Q}+\theta \mathcal{Q}+\pi \mathcal{Q}+w\mathcal{Q},\hfill & \mathrm{if}p\in N^2,\hfill \\ \mathcal{Q}+\theta \mathcal{Q}+\pi \mathcal{Q},\hfill & \mathrm{if}p\in N\setminus N^2.\hfill \end{array}\right.$$

(6)

(iii)

If $n=3,$ we have $\mathcal{Q}=GR(p^3,m).$ The ring R can be expressed as

$$R=\mathcal{Q}+\theta \mathcal{Q}+\pi \mathcal{Q}.$$

(7)

For the purpose of simplicity, let $\xi$ denote a primitive element of $\mathcal{Q};$ we set

$$\begin{array}{cc}\hfill & \mathrm{\Delta }\left(m\right)=\langle \xi \rangle \cup \left\{0\right\}=\{0,1,\xi ,{\xi }^2,\dots ,{\xi }^{p^m-2}\};\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & A=\{{\xi }^{2i+1}:1\le i\le {\displaystyle \frac{p^m-1}{2}}-1\}=\{\alpha \in \mathrm{\Delta }*\left(m\right):\alpha \notin {(\mathrm{\Delta }*\left(m\right))}^2\};\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & B=\{{\xi }^{2i}:1\le i\le {\displaystyle \frac{p^m-1}{2}}-1\}=\{\alpha \in \mathrm{\Delta }*\left(m\right):\alpha \in {(\mathrm{\Delta }*\left(m\right))}^2\}.\hfill \end{array}$$

(10)

As we know, there are ${\gamma }_1$ and ${\gamma }_2\in \mathrm{\Delta }\left(m\right)$ such that

$$\xi ={\gamma }_1^2+{\gamma }_2^2.$$

(11)

3.1. Frobenius Local Rings of Characteristic p

For the duration of this discussion, we have $n=1.$ The primary subring of R is represented as $\mathcal{Q}={\mathbb{F}}_{p^m}.$ The order of R explains that k must equal 4 by Equation (5). So, we divide our discussion into two subcases: (i) when $\theta \pi =0$ and (ii) when $\theta \pi \ne 0.$ To begin, we introduce the following helpful lemmas treating the two cases.

Lemma 2.

Suppose that R is a 5-length local ring with invariants $p,n=1,m,5,k.$ Assuming also that $N^2=\langle {\theta }^2\rangle ,$

$$\theta \pi =\theta w=\pi w=0,$$

then only one of the following holds

$$\left\{\begin{array}{c}\left(\mathrm{i}\right){\pi }^2={\theta }^2\mathit{and}{w}^2=0;\hfill \\ \left(ii\right){\theta }^2=\xi {\pi }^2\mathit{and}{w}^2=0;\hfill \\ \left(iii\right){\theta }^2={\pi }^2\mathit{and}{w}^2={\theta }^2;\hfill \\ \left(iv\right){\theta }^2=\xi {\pi }^2\mathit{and}{w}^2={\theta }^2;\hfill \end{array}\right.$$

where $\xi \in \mathfrak{M}.$

Proof. 

Since $N^2=\langle {\theta }^2\rangle ,$ then

$${\pi }^2={\beta }_1{\theta }^2\mathrm{and}{w}^2={\beta }_2{\theta }^2,$$

(12)

where ${\beta }_1,{\beta }_2\in \mathrm{\Delta }\left(m\right).$ If ${\beta }_2=0$, suppose first that ${\beta }_2=0$ and ${\beta }_1\in B.$ Then, Equation (12) can be reduced to (i) which is ${\pi }^2={\theta }^2$ and $w^2=0$ by $\pi ⟼\sqrt{{\beta }_1^{-1}}\pi ,$$\theta ⟼\theta$, and $w⟼w.$ Note that the same can be said if ${\beta }_2\in B$ and ${\beta }_1=0$ with correspondence $\theta ⟼\theta ,$$\pi ⟼\pi$, and $w⟼\sqrt{{\beta }_2^{-1}}w,$ and thus the relations in Equation (12) are equivalent to $w^2={\theta }^2$ and ${\pi }^2=0.$ Observe that this again is equivalent to (i) by $\theta ⟼\theta$$\pi ⟼w$, and $w⟼\pi .$ Let us assume now that ${\beta }_1\ne$ and ${\beta }_2\ne 0.$ In the case of ${\beta }_1,{\beta }_2\in B,$ Equation (12) becomes ${\pi }^2={\theta }^2$ and $w^2={\theta }^2$ which is (iii) using the correspondence $\theta ⟼\theta ,$$\pi ⟼\sqrt{{\beta }_1^{-1}}\pi$, and $w⟼\sqrt{{\beta }_2^{-1}}w.$ If only one of ${\beta }_1$ or ${\beta }_2$ is in $\mathfrak{M},$ then ${\pi }^2={\beta }_1{\theta }^2\mathrm{and}{w}^2={\beta }_2{\theta }^2$ can be reduced to ${\theta }^2-\xi {\pi }^2=0,w^2-{\theta }^2=0$ (iv) using the correspondence $w⟼\sqrt{{\beta }_2^{-1}}w,$$\pi ⟼\pi$, and $\theta ⟼\theta ,$ where $\xi ={\beta }_1.$ Also a similar argument can take place when ${\beta }_2\in A,$ and using the map $\theta ⟼\theta ,$$\pi ⟼w$, and $w⟼\pi$ will produce the same relations (iv). If both ${\beta }_1\ne {\beta }_2$ are in $A,$ then ${\beta }_1^{-1}={\xi }^{2i+1}$ and ${\beta }_2^{-1}={\xi }^{2j+1}$ for some i and $j.$ The map that takes $\theta ⟼\theta ,$$\pi ⟼{\xi }^{i+j+1}w$, and $w⟼{\xi }^j\pi$ is an isomorphism, and thus Equation (12) turns into ${\theta }^2=\xi {\pi }^2$ and $w^2={\theta }^2$ (iv). In case of ${\beta }_1={\beta }_2,$ and without loss of generality, we let ${\beta }_1={\beta }_2=\xi .$ Assuming $\theta ⟼\theta ,$$\pi ⟼{\xi }^{-1}({\gamma }_1\pi +{\gamma }_{2}w)$, and $w⟼(-{\gamma }_1\pi +{\gamma }_{2}w),$ where ${\gamma }_1^2+{\gamma }_2^2=\xi ,$ then Equation (12) will take the form of (iv). Finally, we have only to show that ${\theta }^2=\xi {\pi }^2$ is equivalent to ${\theta }^2={\pi }^2$ if and only if $p=2.$ Indeed, one can check this fact. Suppose that they are isomorphic. Then there are ${\gamma }_1,{\gamma }_2,{\gamma }_3,{\gamma }_4\in \mathrm{\Delta }\left(m\right)$ such that $u_1={\gamma }_1\theta +{\gamma }_2\pi$ and $v_1={\gamma }_3\theta +{\gamma }_4\pi$ are new generators of N instead of $\theta$ and $\pi .$ These generators must satisfy the relation ${\theta }^2-{\beta }_1{\pi }^2,$ which results in equations of the form ${\gamma }_1^2+{\gamma }_2^2={\beta }_1({\gamma }_3^2+{\gamma }_4^2),$ where ${\gamma }_3^2+{\gamma }_4^2\ne 0$ and ${\gamma }_1{\gamma }_3+{\gamma }_2{\gamma }_4=0$ if and only if $p=2.$ □

Lemma 3.

Let $N=\langle \theta ,\pi ,w\rangle$ and $N^2=\langle \theta \pi \rangle .$ Then, only one of the relations holds

$$\left\{\begin{array}{c}\left\{\begin{array}{c}\left(i\right){\theta }^2={\pi }^2,w^2=0,\theta \pi =0,\hfill \\ \left(\mathit{ii}\right){\theta }^2={\pi }^2,w^2={\theta }^2,\theta \pi =0,\hfill \end{array}\right.\mathit{if}\lambda \in B,\hfill \\ \\ \left\{\begin{array}{c}\left(\mathit{iii}\right){\theta }^2=\xi {\pi }^2,w^2=0,\theta \pi =0,\hfill \\ \left(\mathit{iv}\right){\theta }^2=\xi {\pi }^2,w^2={\theta }^2,\theta \pi =0,\hfill \end{array}\right.\mathit{if}\lambda \in A,\hfill \end{array}\right.$$

where $\xi \in A$ and λ is an element of ${\mathrm{\Delta }}^{*}\left(m\right).$

Proof. 

As $N^2=\langle \theta \pi \rangle ,$ which means that $\theta \pi \ne 0,$ we have

$${\theta }^2={\beta }_1\theta \pi ,{\pi }^2={\beta }_2\theta \pi \mathrm{and}{w}^2={\beta }_3\theta \pi ,$$

(13)

where ${\beta }_1,{\beta }_2,{\beta }_3\in \mathrm{\Delta }\left(m\right).$ Note that if $\theta w\ne 0$ and $\pi w=0,$ then $\theta \pi =\delta \theta w,$ and then by $\theta ⟼\theta ,$$\pi ⟼\pi$, and $w⟼\pi -\delta w,$ we obtain $\theta w=0.$ Similarly when $\pi w\ne 0$ and $\theta w=0,$ if $\theta \pi =\delta \pi w$, we obtain $\pi w=0$ by $\theta ⟼\theta ,$$\pi ⟼\pi$, and $w⟼\theta -\delta w.$ We are left with $\theta w\ne 0$ and $\pi w\ne 0,$ that is, $\theta \pi ={\delta }_1\theta w$ and $\theta \pi ={\delta }_2\pi w$ for some ${\delta }_1,{\delta }_2\in {\mathrm{\Delta }}^{*}\left(m\right).$ If we replace w by $\pi -{\delta }_{1}w,$ then we obtain $\theta w=0,$ and therefore we return to the previous situation with $\theta \pi =\delta \pi w,$ where $\delta =-{\delta }_2{(1-{\delta }_2{\delta }_1^{-1}{\beta }_2)}^{-1}.$ Thus, we proceed with the proof with the assumption that $\theta w=\pi w=0.$ Denote $\lambda$ as

$$\lambda ={\beta }_1{\beta }_2-1.$$

(14)

First, we see that $\lambda \ne 0$ or it will lead to $heta\pi =0$, which is impossible since $N^2\ne 0.$ Let us assume that $\lambda \ne -1,$ which means that ${\beta }_1{\beta }_2\ne 0.$ Now, if we let $\theta ⟼\theta ,$$\pi ⟼\theta -{\beta }_1\pi$, and $w⟼w,$ then direct calculations will force Equation (13) to result in

$${\theta }^2=\xi {\pi }^2,w^2=\delta {\theta }^2\mathrm{and}\theta \pi =0,$$

(15)

where $\xi ={\lambda }^{-1}$ and $\delta ={\beta }_3{\beta }_1^{-1}.$ When ${\beta }_3=0,$ then Equation (15) turns into (i) or (iii), depending on whether $\lambda \in B$ or $\lambda \in A,$ respectively. On the other hand, if ${\beta }_3\ne 0,$ then the relations in Equation (15) are reduced to either (ii) or (iv) based on the fact that $\lambda \in B$ or $\lambda \in A,$ respectively. In fact, the case when $\lambda \in B$ implies that $\xi \in B,$ and hence, by $\theta ⟼\theta ,$$\pi ⟼\sqrt{\xi }\pi$, and $w⟼w,$, Equation (15) corresponds to (ii) if $\delta \in B,$ and to (iv) when $\delta \in A$ with a small adjustment by $\theta ⟼\theta ,$$\pi ⟼w$, and $w⟼\pi .$ Now, suppose that $\lambda =-1$, which we obtain if ${\beta }_1=0,$${\beta }_2=0$, or both. We consider two cases: (a) $\lambda \in B;$ (b) $\lambda \in A.$ Let $\lambda \in B;$ then Equation (15) will be

$$\left\{\begin{array}{c}\left(\mathrm{i}\right),\mathrm{if}\delta =0,\hfill \\ \left(\mathrm{ii}\right),\mathrm{if}\delta \in B,\hfill \\ \left(\mathrm{iv}\right),\mathrm{if}\delta \in A.\hfill \end{array}\right.$$

The other case is when $\lambda \in \mathfrak{M};$ therefore, Equation (15) takes the form of

$$\left\{\begin{array}{c}\left(\mathrm{iii}\right),\mathrm{if}\delta =0,\hfill \\ \left(\mathrm{iv}\right),\mathrm{otherwise}.\hfill \end{array}\right.$$

Note that if we have $\delta \in A,$ we return to a similar case as Lemma 3, and hence we only obtain (iii) or (iv). Finally, we deal with the situation when ${\beta }_1={\beta }_2=0$ and ${\beta }_3\ne 0,$ i.e., $w^2={\beta }_3\theta \pi .$ If we replace $\theta$ by ${\beta }_3\theta ,$$\pi$ by $\pi$, and w by $w,$ we obtain $w^2=\theta \pi .$ As ${\theta }^2-{\pi }^2=0,$ then we let $\theta ⟼\theta +\pi ,$$\pi ⟼\theta -\pi$, and $w⟼w,$ and hence ${\theta }^2=2w^2,$${\pi }^2=-2w^2$, and $\theta \pi =0.$ This case has been dealt with in Lemma 2. If $p=2,$ then the same lemma concludes that we have (iv). If $p\ne 2,$ then we obtain (ii) or (iv) according to whether $\lambda \in B$ or $\lambda \in B,$ respectively, by letting $\theta ⟼w,$$\pi ⟼\pi$, and $w⟼\theta .$ □

Remark 2.

In light of Lemma 3, we observe that when $p=2,$ the relation ${\theta }^2={\pi }^2=0$ is not equivalent with ${\theta }^2-{\pi }^2=0.$ Thus, as we see later, $\langle {\theta }^2,{\pi }^2\rangle \ne \langle {\theta }^2-{\pi }^2,\theta \pi \rangle .$

Theorem 1.

Let R be a Frobenius local ring with invariants $p,n=1,m,5,k.$ Then, R is isomorphic to a unique copy of the following:

(i) 

If $p\ne 2,$

$$\begin{array}{cccc}{\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^3,{\pi }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-{\pi }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-\xi {\pi }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},& \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-{\pi }^2,{\theta }^2-w^2,\theta \pi ,\theta w,\pi w\rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-\xi {\pi }^2,{\theta }^2-w^2,\theta \pi ,\theta w,\pi w\rangle }}.& \end{array}$$

(ii) 

If $p=2,$

$$\begin{array}{cccc}{\displaystyle \frac{{\mathbb{F}}_{2^m}[\theta ,\pi ,w]}{\langle {\theta }^3,{\pi }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},& {\displaystyle \frac{{\mathbb{F}}_{2^m}[\theta ,\pi ,w]}{\langle {\pi }^2-{\theta }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},& {\displaystyle \frac{{\mathbb{F}}_{2^m}[\theta ,\pi ,w]}{\langle {\pi }^2-{\theta }^2,w^2-{\theta }^2,\theta \pi ,\theta w,\pi w\rangle }},& {\displaystyle \frac{{\mathbb{F}}_{p{2}^m}[\theta ,\pi ,w]}{\langle {\theta }^2,{\pi }^2,w^2,\theta w,\pi w\rangle }}.\end{array}$$

Proof. 

We make use of Equation (4); $\vartheta \left(N\right)+\vartheta \left(N^2\right)\ge k+1.$ Since $n=1,$ then $k=4$ since $\vartheta \left(N^2\right)\le \vartheta {\left(N\right)}^2.$ Also, because $n=1,$ we have $\mathcal{Q}\subset R,$ and since $t=3,$ then

$$R=\mathcal{Q}+\theta \mathcal{Q}+\pi \mathcal{Q}+w\mathcal{Q}+\zeta \mathcal{Q}.$$

Furthermore, $N=\langle \theta ,\pi ,w,\zeta \rangle$ is

$$N=\theta \mathcal{Q}+\pi \mathcal{Q}+w\mathcal{Q}+\zeta \mathcal{Q}.$$

The minimal generator elements $\theta ,\pi ,w$, and $\theta$ over $\mathcal{Q}$ satisfy either $\vartheta \left(N\right)=3$ and $\vartheta \left(N^2\right)=1$ or $\vartheta \left(N\right)=2$ and $\vartheta \left(N^2\right)=2.$ As R is Frobenius, then we must have $\vartheta \left(N^2\right)=1.$ Note that all multiplications of $\theta ,\pi ,w$ are in $N^2.$ Suppose first that ann$\left(\theta \right)=\langle {\theta }^2\rangle ,$ and hence $N^2=\langle {\theta }^2\rangle =\langle w\rangle ,$ and so

$${\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^3,{\pi }^2,w^2,\theta \pi ,\pi w,\theta w\rangle }}.$$

Such a ring is uniquely determined. On the other hand, if ann$\left(\theta \right)=\langle \pi \rangle$ or ann$\left(\theta \right)=\langle w\rangle$, which means $N^2=\langle {\pi }^2\rangle =\langle {\theta }^2\rangle$ or $N^2=\langle w^2\rangle =\langle {\theta }^2\rangle$, then we obtain

$${\pi }^2={\beta }_1{\theta }^2,w^2={\beta }_2{\theta }^2\mathrm{and}\theta \pi =\pi w=\theta w=0.$$

Hence,

$$R\cong {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-{\beta }_1{\pi }^2,{\theta }^2-{\beta }_2{w}^2,\theta \pi ,\pi w,\theta w\rangle }},$$

where ${\beta }_1,{\beta }_2\in \mathrm{\Delta }\left(m\right).$ Now, by Lemma 2, we obtain

$$R\cong \left\{\begin{array}{cc}{\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\pi }^2-{\theta }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},\hfill & \mathrm{if}\left(\mathrm{i}\right)\mathrm{holds},\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-\xi {\pi }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},\hfill & \mathrm{if}\left(\mathrm{ii}\right)\mathrm{holds},\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-{\pi }^2,{\theta }^2-w^2,\theta \pi ,\theta w,\pi w\rangle }}\hfill & \mathrm{if}\left(\mathrm{iii}\right)\mathrm{holds},\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-\xi {\pi }^2,{\theta }^2-w^2,\theta \pi ,\theta w,\pi w\rangle }}\hfill & \mathrm{if}\left(\mathrm{iv}\right)\mathrm{holds}.\hfill \end{array}\right.$$

(16)

Finally, without loss of generality, we consider $N^2=\langle \theta \pi \rangle ;$ in such a case, ann$\left(\theta \right)=\langle \theta -\beta \pi \rangle ,$ where $\beta \in \mathrm{\Delta }*\left(m\right).$ The relations, therefore, hold

$${\theta }^2-{\beta }_1\theta \pi =0,{\pi }^2-{\beta }_2\mathrm{and}{w}^2-{\beta }_3\theta \pi =0,$$

for some ${\beta }_1,{\beta }_2$, and ${\beta }_3$ in $\mathrm{\Delta }\left(m\right).$ Thus,

$$R={\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-{\beta }_1\theta \pi ,{\pi }^2-{\beta }_2\theta \pi ,w^2-{\beta }_3\theta \pi ,\pi w,\theta w\rangle }}.$$

In light of Lemma 3, we have R structured as in Equation (16). For $p=2,$ there is an additional ring which is ${\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2,{\pi }^2,w^2,\pi w,\theta w\rangle }}.$ As elements of ${\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2,{\pi }^2,w^2,\pi w,\theta w\rangle }}$ have square zero, and also $\theta$ and $\pi$ of ${\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-{\pi }^2,w^2,\theta \pi ,\pi w,\theta w\rangle }}$ such that ${\theta }^2={\pi }^2\ne 0,$ therefore, ${\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2,{\pi }^2,w^2,\pi w,\theta w\rangle }}\neg \cong {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-{\pi }^2,w^2,\theta \pi ,\pi w,\theta w\rangle }}.$ To sum up, for $p\ne 2$, there are 5 rings and for $p=2$, there are 4 rings. □

3.2. Frobenius Local Rings of Characteristic $p^2$

For the purpose of this discussion, let R be a Frobenius local ring with characteristic $p^2$, which indicates that $n=2$. Thus, $\mathcal{Q}=GR(p^2,m)$. Proposition 1 states that k can be either 3 or 2. Therefore, we will divide our discussion into two subcases: (i) when $p\in \mathrm{ann}\left(N\right)$ and (ii) when $p\notin \mathrm{ann}\left(N\right)$. To start, we present the following useful lemma investigating the first case, while the second case is discussed later.

Lemma 4.

Let R be a local ring with invariants $p,n=2,m,5,k.$ Assuming $N=\langle \theta ,\pi ,w\rangle$ and $N^2=\langle p\rangle ,$ then we obtain

$$\left\{\begin{array}{c}{\theta }^2={\delta }_{1}p,\hfill \\ {\pi }^2={\delta }_2{\theta }^2,\hfill \\ w^2={\delta }_3{\theta }^2,\hfill \end{array}\right.$$

$$({\delta }_1,{\delta }_2,{\delta }_3)\in \{(1,0,0),(1,1,0),(1,1,1),(\xi ,0,0),(\xi ,1,0),(\xi ,1,1,)\},$$

where $\xi \in \mathfrak{M}.$

Proof. 

Because $N^2=\langle p\rangle ,$ then

$${\theta }^2=p{\beta }_1,{\pi }^2=p{\beta }_2\mathrm{and}{w}^2=p{\beta }_3,$$

(17)

where ${\beta }_1,{\beta }_2,{\beta }_3\in \mathrm{\Delta }\left(m\right).$ First, we note that if, without loss of generality, $\theta \pi \ne 0,$ then we can shift all relations of Equation (17) to those of Equation (13) in the proof of Lemma 3. Hence, we continue this argument with $\theta \pi =\theta w=\pi w=0.$ Note that we cannot have all ${\beta }_i^{\prime }$s to be zeros; otherwise, $N^2=0.$ If we fix ${\theta }^2={\beta }_{1}p,$ then we obtain

$${\theta }^2={\delta }_{1}p,{\pi }^2={\delta }_2{\theta }^2\mathrm{and}{w}^2={\delta }_3{\theta }^2,$$

(18)

where ${\delta }_1={\beta }_1,$${\delta }_2={\beta }_1^{-1}{\beta }_2$, and ${\delta }_2={\beta }_1^{-1}{\beta }_3.$ If ${\delta }_1\in B,$ then there are two options for ${\delta }_2$ which are $\delta =0$ or $\delta \ne 0.$ The first case gives $({\delta }_1,{\delta }_2,\delta )$ as one of $(1,0,0),$ $(1,0,1)$, or $(1,0,\xi ).$ However, $(1,0,1)\cong (1,1,0)$ by $\theta ⟼\theta ,$$\pi ⟼w$, and $w⟼\pi .$ Also $(1,0,\xi )\cong (\xi ,0,1)\cong (\xi ,1,0)$ by $\theta ⟼w,$$\pi ⟼\pi$, and $w⟼\theta$, followed by $\theta ⟼\theta ,$$\pi ⟼w$, and $w⟼\pi .$ In the second case, when ${\delta }_2\ne 0,$ we have $({\delta }_1,{\delta }_2,\delta )$ as $(1,1,0),(1,1,1),(1,1,\xi ),(1,\xi ,0),(1,\xi ,1)$, or $(1,\xi ,{\alpha }_1),$ where ${\alpha }_1\in A.$ By a similar argument, one can note that $(1,\xi ,0)\cong (\xi ,1,0)$ and $(1,1,\xi )\cong (\xi ,1,1).$ Moreover, we observe that $(1,\xi ,{\alpha }_1)\cong (1,\xi ,1)\cong (\xi ,1,1)$ from Lemma 2. On the other hand, if $\xi ={\delta }_1\in \mathfrak{M},$ we have $(\xi ,0,0),(\xi ,1,0),(\xi ,1,1),(\xi ,{\alpha }_1,0),(\xi ,{\alpha }_1,1),(\xi ,{\alpha }_1,{\alpha }_2),(\xi ,0,1),(\xi ,0,{\alpha }_2)$, and $(\xi ,1,{\alpha }_2).$ Note that $(\xi ,1,0)\cong (\xi ,0,1),$$(\xi ,{\alpha }_1,0)\cong (\xi ,0,{\alpha }_2)$, and $(\xi ,1,{\alpha }_2)\cong (\xi ,{\alpha }_1,1).$ If we replace $\theta ⟼\theta ,$$\pi ⟼\sqrt{{\left(\xi {\alpha }_1\right)}^{-1}}\pi$, and $w⟼w,$ then $(\xi ,{\alpha }_1,0)\cong (1,\xi ,0)$ and $(\xi ,{\alpha }_1,1)\cong (1,\xi ,1)$, which is, in turn, equivalent to $(\xi ,1,0)$ and $(\xi ,1,1),$ respectively. Finally, $(\xi ,{\alpha }_1,{\alpha }_2)\cong (1,\xi ,1)\cong (\xi ,1,1)$ by assuming $\theta ⟼\theta ,$$\pi ⟼\sqrt{{\left(\xi {\alpha }_1\right)}^{-1}}\pi$, and $w⟼\sqrt{{\left(\xi {\alpha }_2\right)}^{-1}}w$, followed by $\theta ⟼\pi ,$$\pi ⟼\theta$, and $w⟼w.$ □

Theorem 2.

Suppose R is a Frobenius local ring with invariants $p,n=2,m,5,k.$ Then, R is isomorphic to a unique copy of the following:

(i) 

If $p\ne 2,$

$$\begin{array}{ccc}{\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-p,{\pi }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-p,{\pi }^2-{\theta }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-p,{\pi }^2-{\theta }^2,w^2-{\theta }^2,\theta \pi ,\theta w,\pi w\rangle }},\\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-\xi p,{\pi }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-p\xi ,{\pi }^2-{\theta }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-p\xi ,{\pi }^2-{\theta }^2,w^2-{\theta }^2,\theta \pi ,\theta w,\pi w\rangle }},\\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2,\theta \pi ,p\pi \rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2-p\theta ,\theta \pi ,p\pi \rangle }}.\end{array}$$

(ii) 

If $p=2,$

$$\begin{array}{ccc}{\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-2,{\pi }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-2,{\pi }^2-{\theta }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-2,{\pi }^2-{\theta }^2,w^2-{\theta }^2,\theta \pi ,\theta w,\pi w\rangle }},\\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2,{\pi }^2,w^2,\theta \pi -2,\theta w,\pi w\rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2,\theta \pi ,2\pi \rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2-2\theta ,\theta \pi ,2\pi \rangle }},{\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2-2\theta ,{\pi }^2,\theta \pi ,2\pi \rangle }}.\end{array}$$

Proof. 

We proceed with the two cases. (Case a.) If $p\in N^2=$ ann$\left(N\right),$ then $N^2=\langle p\rangle$ since R is Frobenius. The size of R gives $k=3$ and

$$R=\mathcal{Q}+\theta \mathcal{Q}+\pi \mathcal{Q}+w\mathcal{Q},$$

$$N=\theta \mathcal{Q}+\pi \mathcal{Q}+w\mathcal{Q}.$$

The generators of N hold the relations,

$${\theta }^2=p{\beta }_1,{\pi }^2=p{\beta }_2\mathrm{and}{w}^2=p{\beta }_3,$$

(19)

where ${\beta }_1,{\beta }_2,{\beta }_3\in \mathrm{\Delta }\left(m\right).$ As in the proof of Lemma 4, we have $\theta \pi =\theta w=\pi w=0.$ Thus, the construction of R is given by

$$R={\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-p{\beta }_1,{\pi }^2-p{\beta }_2,w^2-p{\beta }_3,\theta \pi ,\pi w,\theta w\rangle }}.$$

By Lemma 4, when $p\ne 2,$

$$R\cong \left\{\begin{array}{cc}{\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-p,{\pi }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},\hfill & \mathrm{if}({\delta }_1,{\delta }_2,{\delta }_3)=(1,0,0),\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-p,{\pi }^2-{\theta }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},\hfill & \mathrm{if}({\delta }_1,{\delta }_2,{\delta }_3)=(1,1,0),\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-p,{\pi }^2-{\theta }^2,w^2-{\theta }^2,\theta \pi ,\theta w,\pi w\rangle }},\hfill & \mathrm{if}({\delta }_1,{\delta }_2,{\delta }_3)=(1,1,1),\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-p\xi ,{\pi }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},\hfill & \mathrm{if}({\delta }_1,{\delta }_2,{\delta }_3)=(\xi ,0,0),\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-p\xi ,{\pi }^2-{\theta }^2,w^2,\theta \pi ,\theta w,\pi w\rangle }},\hfill & \mathrm{if}({\delta }_1,{\delta }_2,{\delta }_3)=(\xi ,1,0),\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2-p\xi ,{\pi }^2-{\theta }^2,w^2-{\theta }^2,\theta \pi ,\theta w,\pi w\rangle }},\hfill & \mathrm{if}({\delta }_1,{\delta }_2,{\delta }_3)=(\xi ,1,1).\hfill \end{array}\right.$$

(20)

When $p=2,$ we have $(2,2^m-1)=1,$ and thus ${\mathrm{\Delta }}^{*}\left(m\right)=B.$ Therefore, we only have $(1,0,0),(1,1,0)$, and $(1,1,1)$ which give 3 different rings. Moreover, the ring ${\displaystyle \frac{\mathcal{Q}[\theta ,\pi ,w]}{\langle {\theta }^2,{\pi }^2,w^2,\theta \pi -2,\theta w,\pi w\rangle }}$ is not isomorphic to any ring associated with $(1,0,0),(1,1,0)$, and $(1,1,1);$ please see Remark 2. (Case b.) Let $p\in N\setminus N^2,$ and since R is Frobenius, then $p\notin$ ann$\left(N\right)$, which means that $p\theta \ne 0.$ In this case, $k=2,$ and thus

$$R=\mathcal{Q}+\theta \mathcal{Q}+\pi \mathcal{Q},$$

$$N=p\mathcal{Q}+\theta \mathcal{Q}+\pi \mathcal{Q},$$

$$N^2=p\theta \mathcal{Q}.$$

By the order of $R,$$p\pi =0.$ Moreover, we have

$${\theta }^2=p{\beta }_1\theta ,{\pi }^2=p{\beta }_2\theta \mathrm{and}\theta \pi =p{\beta }_3\theta .$$

(21)

Replacing $\theta ⟼\theta$ and $\pi ⟼\pi -p{\beta }_3$ gives $\theta \pi =0.$ If $p\ne 2,$ completing squares gives ${\theta }^2=0.$ Therefore, we have ${\pi }^2=0,$${\pi }^2=p\theta$, or ${\pi }^2=p\xi \theta$, depending on ${\beta }_2=0,{\beta }_2\in B$, or ${\beta }_2\in \mathfrak{M},$ respectively. But since we can use the correspondence $\theta ⟼\xi \theta$ and $\pi ⟼\pi ,$ then we may take ${\beta }_2$ to be 0 or $1.$ Hence, there are 2 rings which are

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2,\theta \pi \rangle }},\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2-p\theta ,\theta \pi \rangle }}.\hfill \end{array}\right.$$

When $p=2,$ we get ${\theta }^2=2\theta$ or ${\theta }^2=0,$ and hence, we obtain 3 such rings,

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2,\theta \pi \rangle }},\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2-2\theta ,\theta \pi \rangle }},\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2-2\theta ,{\pi }^2,\theta \pi \rangle }}.\hfill \end{array}\right.$$

□

3.3. Frobenius Local Rings of Characteristic $p^3$

This subsection addresses the scenario where R is a Frobenius local ring with characteristic $p^3$, signifying that $n=3.$ The primary subring of R is expressed as $\mathcal{Q}=GR(p^3,m)$. According to Proposition 1, k takes the value 2. In this instance, we find ourselves in a situation similar to that described in Lemmas 3 and 4; therefore, we will move directly to the subsequent result without presenting an analogous lemma.

Theorem 3.

Suppose R is a Frobenius local ring with invariants $p,n=3,m,5,k.$ Then, R is isomorphic to a unique copy of the following:

(i) 

If $p\ne 2,$

$$\begin{array}{ccc}{\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2,\theta \pi ,p\theta ,p\pi \rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2-p^2,{\pi }^2,\theta \pi ,p\theta ,p\pi \rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2-p^2\xi ,{\pi }^2,\theta \pi ,p\theta ,p\pi \rangle }},\\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2-p^2,{\pi }^2-p^2,\theta \pi ,p\theta ,p\pi \rangle }},& {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2-p^2,{\pi }^2-p^2\xi ,\theta \pi ,pheta,p\pi \rangle }}.\end{array}$$

(ii) 

If $p=2,$

$$\begin{array}{cccc}{\displaystyle \frac{GR(8,m)[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2,\theta \pi ,2\theta ,2\pi \rangle }},& {\displaystyle \frac{GR(8,m)[\theta ,\pi ]}{\langle {\theta }^2-4,{\pi }^2,\theta \pi ,2\theta ,2\pi \rangle }},& {\displaystyle \frac{GR(8,m)[\theta ,\pi ]}{\langle {\theta }^2-4,{\pi }^2-4,\theta \pi ,2\theta ,2\pi \rangle }},& {\displaystyle \frac{GR(8,m)[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2,\theta \pi -4,2\theta ,2\pi \rangle }}.\end{array}$$

Proof. 

As $n=3=t,$ then p must be in $N\setminus N^2;$ otherwise, $p^2\in N^4=0$, which is impossible. Thus, $N^2=\langle p^2\rangle ,$ and hence k must be $2.$ If $k=1,$ then $0\ne p\theta =p^2\beta ,$ and by $\theta ⟼\theta -p\beta ,$ we obtain $p\theta =0$, which contradicts the order of $R.$ So,

$$R=\mathcal{Q}+\theta \mathcal{Q}+\pi \mathcal{Q},$$

$$N=\mathcal{Q}+\theta \mathcal{Q}+\pi \mathcal{Q},$$

$$N^2=\mathcal{Q}.$$

The generators $\theta ,\pi$ have the following relations

$${\theta }^2=p^2{\beta }_1,{\pi }^2=p^2{\beta }_2,$$

where ${\beta }_1$ and ${\beta }_2$ are taken from $\mathrm{\Delta }\left(m\right).$ First, we note that if $p\ne 2,$ we assume $\theta \pi =0,$ because otherwise we would have a situation similar to that in Lemma 3. By the same argument as in Lemma 2 (by treating ${\theta }^2$ as $p^2$ and $\theta ,\pi$ as $\pi ,w$), we obtain

$$R\cong \left\{\begin{array}{cc}{\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2,\theta \pi ,p\theta ,p\pi \rangle }},\hfill & \mathrm{if}{\beta }_1={\beta }_2=0,\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2-p^2,{\pi }^2,\theta \pi ,p\theta ,p\pi \rangle }},\hfill & \mathrm{if}\left(\mathrm{i}\right)\mathrm{holds},\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2-p^2\xi ,{\pi }^2,\theta \pi ,p\theta ,p\pi \rangle }},\hfill & \mathrm{if}\left(\mathrm{ii}\right)\mathrm{holds},\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2-p^2,{\pi }^2-p^2,\theta \pi ,p\theta ,p\pi \rangle }},\hfill & \mathrm{if}\left(\mathrm{iii}\right)\mathrm{holds},\hfill \\ {\displaystyle \frac{\mathcal{Q}[\theta ,\pi ]}{\langle {\theta }^2-p^2\xi ,{\pi }^2-p^2,\theta \pi ,p\theta ,p\pi \rangle }},\hfill & \mathrm{if}\left(\mathrm{iv}\right)\mathrm{holds}.\hfill \end{array}\right.$$

When $p=2,$ we have a case, as we observed before, of ${\displaystyle \frac{GR(8,m)[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2,\theta \pi -4,2\theta ,2\pi \rangle }}\neg \cong {\displaystyle \frac{GR(8,m)[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2,\theta \pi ,2\theta ,2\pi \rangle }}.$ □

Corollary 1.

Let $m=1,$ i.e., rings with residue field ${\mathbb{F}}_p.$ Then, the number is

$$N_F=\left\{\begin{array}{cc}16,\hfill & \mathit{if}p=2,\hfill \\ 17,\hfill & \mathit{if}p\ne 2.\hfill \end{array}\right.$$

This example demonstrates how the considered local rings in this paper can be employed in the construction of linear codes. Linear codes of length s over R are R-submodules of $R^s$; specifically, cyclic codes of length s over R are just ideals of the quotient ring ${\displaystyle \frac{R\left[x\right]}{\langle x^s-1\rangle }}$. For additional information regarding the background of linear codes over finite rings, we refer to [7,11,15].

Example 1.

Suppose $R={\displaystyle \frac{{\mathbb{Z}}_8[\theta ,\pi ]}{\langle {\theta }^2,{\pi }^2,\theta \pi ,2\theta ,2\pi \rangle }}.$ Then, by Theorem 3, we have that R is a Frobenius local ring of order $p^5$ and $\{\theta ,\pi ,2\}$ is a minimal generating set for N, the maximal ideal of R, $N^2=\langle 4\rangle$. Moreover, we have $\mathrm{\Delta }\left(1\right)=\{0,1\}\subset \mathcal{Q}$, where $\mathcal{Q}={\mathbb{Z}}_8.$ Supposing also that $s=7,$ then $x^7-1=(x+1)(x^3+x+1)(x^3+x^2+1)$ over ${\mathbb{F}}_2,$ and hence, by Hensel’s Lemma, $x^7-1=q_1\left(x\right)q_2\left(x\right)q_3\left(x\right)$ over $R.$ In this case,

$${\displaystyle \frac{R\left[x\right]}{\langle x^7-1\rangle }}\cong R+R_2+R_3,(asdirectsum)$$

by the Chinese Reminder Theorem, where $R_2={\displaystyle \frac{R\left[x\right]}{\langle q_2\left(x\right)\rangle }}$ and $R_3={\displaystyle \frac{R\left[x\right]}{\langle q_3\left(x\right)\rangle }}.$ Set $\mathrm{\Delta }\left(3\right)=\{{\alpha }_0+{\alpha }_{1}x+{\alpha }_2{x}^2:{\alpha }_i\in \mathrm{\Delta }\left(1\right)\}$ for the rings $R_2$ and $R_3.$ Thus, if C is a cyclic code over R of length $7,$ then C is an ideal of ${\displaystyle \frac{R\left[x\right]}{\langle x^7-1\rangle }},$ and can be expressed by the form (as direct sum)

$$C\cong C_1+C_2+C_3,$$

where $C_i$ are ideals of $R,$$R_1$, and $R_3,$ respectively. Now, consider the following two cases:

(i) 

If $C_1=\langle \theta +\beta \pi ,2\rangle ,$$C_2=\langle \theta +{\delta }_1\pi +2{\delta }_2\rangle$, and $C_3=\langle \pi ,2\rangle ,$ where β is taken from $\mathrm{\Delta }\left(1\right)$ and ${\delta }_i$ are in $\mathrm{\Delta }\left(3\right),$ this means that the ideal, in $R+R_2+R_3$, is of the form

$$C\cong \langle \theta +\beta \pi ,2\rangle +\langle \theta +{\delta }_1\pi +2{\delta }_2\rangle +\langle \pi ,2\rangle ,$$

and this is isomorphic to the cyclic code, in ${\displaystyle \frac{R\left[x\right]}{\langle x^7-1\rangle }}$,

$$C=\langle (\theta +\beta \pi ,2)q_2\left(x\right)q_3\left(x\right),(\theta +{\delta }_1\pi +2{\delta }_2)q_1\left(x\right)q_3\left(x\right),\pi q_1\left(x\right)q_2\left(x\right),2q_1\left(x\right)q_2\left(x\right)\rangle .$$

(ii) 

If $C_1=\langle \theta +\beta \pi \rangle ,$$C_2=\langle 4\rangle$, and $C_3=\langle \theta ,\pi ,2\rangle ,$ where $\beta \in {\mathbb{F}}_2,$ then, we obtain that the associated ideal, in $R+R_2+R_3$, has the expression

$$C\cong \langle \theta +\beta \theta \rangle +\langle 4\rangle +\langle \theta ,\pi ,2\rangle .$$

In ${\displaystyle \frac{R\left[x\right]}{\langle x^7-1\rangle }}$, this is isomorphic to

$$C=\langle (\theta +\beta \pi )q_2\left(x\right)q_3\left(x\right),4q_1\left(x\right)q_3\left(x\right),\theta q_1\left(x\right)q_2\left(x\right),\pi q_1\left(x\right)q_2\left(x\right),2q_1\left(x\right)q_2\left(x\right)\rangle .$$

4. Conclusions

The importance of local rings defined by $p,n,m,l,$ and k in coding theory has resulted in their prominence as a study focus. Frobenius local rings have attracted significant attention because of their essential function in distance distributions and error correcting mechanisms. This article examines local rings of length 5 with $t=3$, particularly concentrating on rings of order $p^{5m}$. We effectively list all such rings up to isomorphism, specifying their invariants. Furthermore, we categorize each Frobenius ring satisfying these criteria, as presented in

Table 1. Enumeration of Frobenius local rings of order $p^{5m}$.

Char (R) $={\mathit{p}}^{\mathit{n}}$	Number of Non-Isomorphic Classes
	$\mathit{p}\ne \mathbf{2}$	$\mathit{p}=\mathbf{2}$
$n=1$	4	5
$n=2$ and $p\notin N^2$	2	3
$n=2$ and $p\in N^2$	6	4
$n=3$	5	4

. This classification not only emphasizes the structural diversity of these rings but also highlights their potential applications in coding theory, particularly in enhancing error correction capabilities.