A Class of Local Non-Chain Rings of Order p^5m

Abstract

This study investigates finite commutative local non-chain rings characterized by the well-established invariants $p, n, m, l$, and k, where p denotes a prime number. We specifically focus on Frobenius local rings with length $l=5$ and an index of nilpotency $t=4$. The significance of Frobenius rings in coding theory arises when specific results of linear codes are applicable to both finite fields and finite Frobenius rings. In light of this, we provide a comprehensive classification and enumeration of Frobenius local rings of order $p^{5m}$ with $t=4$, highlighting their distinctive properties in relation to varying values of $n.$ This research advances our understanding of the structural features of Frobenius rings and their applications in coding theory.

1. Introduction

Mathematicians have paid much attention to finite local rings since they are a key class in algebraic structures with certain features. According to [1,2,3], a finite local ring is one in which each ideal is contained within a single maximum ideal. More broad rings might possess a more complicated structure of ideals due to the possibility of having many maximal ideals, which is in stark contrast to this property. Galois rings, initially presented by Krull [4], provide a fundamental illustration of finite local rings. In particular, the basic ideas related to finite local rings are shown by rings of the type ${\mathbb{Z}}_{p^n}$, where p is a prime number. Since finitely produced modules over finite local rings display distinctive traits that provide details about the broader framework of the rings, the investigation of these rings is strongly related to module theory. To obtain the socle of a module $\mathfrak{M}$ over a ring R, we add up all the smallest R-submodules of $\mathfrak{M}$ and call it $\mathrm{soc}\left(\mathfrak{M}\right)$. When R is seen as a R-module, a ring is called Frobenius if and only if $\mathrm{soc}\left(R\right)\cong {\displaystyle \frac{R}{N}}\cong {\mathbb{F}}_{p^m}$. The application of two classical theorems by MacWilliams—the Extension Theorem and the MacWilliams Identities—has led to the recent demonstration that Frobenius local rings can serve as alphabets for linear codes. Similarly to how these theorems are applicable in finite fields, they are also applicable to Frobenius local rings. Refer to [5,6,7,8,9,10] for additional information on this topic.

In this article, we consider finite rings that are commutative and possess an identity element. Each local ring R can be expressed over its prime subring $\mathcal{O}$ as

$$R\cong {\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_k]}{\Im }},$$

(1)

where ℑ is an appropriate ideal in terms of the indeterminates ${\vartheta }_i$ over $\mathcal{O}$ [11]. To see a more explicit description about the structure of $\Im ,$ we refer to our results in Theorems 1–4. We summarize some facts about any local ring $R,$

$$\left\{\begin{array}{c}N={\displaystyle \frac{⟨p,{\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_k⟩}{\Im }};\left(\mathrm{the}\mathrm{maximal}\mathrm{ideal}\mathrm{of}R\right)\hfill \\ {\displaystyle \frac{R}{N}}\cong {\mathbb{F}}_{p^m};\left(\mathrm{the}\mathrm{residue}\mathrm{field}\mathrm{of}R\right)\hfill \\ \left|R\right|={\left|{\mathbb{F}}_{p^m}\right|}^l={\left(p^m\right)}^l;\left(\mathrm{the}\mathrm{order}\mathrm{of}R\right)\hfill \\ N^t=0\mathrm{and}{N}^{t-1}\ne 0;\left(\mathrm{the}\mathrm{index}\mathrm{of}\mathrm{nilpotency}\mathrm{of}N\right)\hfill \\ o_{+}\left(1\right)=p^n,\left(\mathrm{the}\mathrm{characteristic}\mathrm{of}R\right)\hfill \end{array}\right.$$

where $n, m, l$, and t are positive integers, and l is the length of $R.$ Analysis of ℑ structure is the main key of the categorization of Frobenius local rings. A direct sum over their prime subring $\mathcal{O}=GR(p^n,m)$ is a common way to represent such rings, which is why this result follows:

$$R=\mathcal{O}+{\vartheta }_1\mathcal{O}+\cdots +{\vartheta }_k\mathcal{O}$$

(2)

for some positive integer k. In this study, we call the numbers p, n, m, k, and l the invariants of $R.$ The objective of this paper is to identify and classify all Frobenius local rings that possess fixed invariants where $l=5$, particularly focusing on those rings with an index of nilpotency $t=4$.

There is a unique local ring with p elements, up to isomorphism, which is the Galois field ${\mathbb{F}}_p$, while Fobenius local rings of order $p^2$ include ${\mathbb{F}}_{p^2}$, ${\mathbb{Z}}_{p^2}$, and ${\displaystyle \frac{{\mathbb{F}}_p\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2⟩}}$. For odd primes p, the local rings of size $p^3$ consist of ${\mathbb{F}}_{p^3}$, ${\mathbb{Z}}_{p^3}$, ${\displaystyle \frac{{\mathbb{F}}_p\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^3⟩}}$, ${\displaystyle \frac{{\mathbb{Z}}_{p^2}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2-p,p{\vartheta }_1⟩}}$, and ${\displaystyle \frac{{\mathbb{Z}}_{p^2}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2-p\zeta ,p{\vartheta }_1⟩}}$ (where $\zeta$ is a generator of ${\mathbb{F}}_p^{*}$). If $p=2$, we have ${\mathbb{F}}_{2^3}$, ${\mathbb{Z}}_{2^3}$, ${\displaystyle \frac{{\mathbb{F}}_2\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^3,2{\vartheta }_1⟩}}$, and ${\displaystyle \frac{{\mathbb{Z}}_{2^2}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2-2,2{\vartheta }_1⟩}}.$ In [12], a partial examination of local rings was conducted when $l=5$ and $m=1.$ Additionally, such rings of size $p^4$ were investigated in [9], as linear codes over these rings were described. The study of finite Frobenius local rings containing $p^{5m}$ elements for $m>1$ offers a compelling avenue for additional investigation. The present article aims to elucidate the recognition of Frobenius local rings characterized by a length of 5 and an index of nilpotency of 4.

Here is the structure of the study: We lay up the groundwork for understanding finite local rings and the modules that go along with them in Section 2. Then, in Section 3, we determine all Frobenius local rings with $p^{5m}$ elements, where $t=4$, and p is a prime integer, and we outline our extensive analysis. For n values of $1,$ 2, 3, or 4, this section is further broken into subsections that deal with those particular values.

2. Preliminaries

In this section, we define rings R and $\mathfrak{N}$, where R is a subring of $\mathfrak{N}$. The ideal $IT$ refers to the extension of the ideal ℑ from R to $\mathfrak{N}$. For any R-module $\mathfrak{M}$, the annihilator ideal is given by

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

We can naturally turn $\mathfrak{M}$ into a module over the quotient ${\displaystyle \frac{R}{\Im }}$ for any ideal ℑ of R that lies within ${\mathrm{Ann}}_R\left(\mathfrak{M}\right)$. The multiplication defined is $(r+\Im )a=ra,$ for $r\in R$ and $a\in \mathfrak{M}$, ensuring the corresponding submodules of $\mathfrak{M}$ over R and ${\displaystyle \frac{R}{\Im }}$ align consistently. A chain of strict inclusions of R-submodules of $\mathfrak{M}$ can be expressed as

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

where l denotes the number of inclusions. This series is termed a composition series if each quotient ${\displaystyle \frac{{\mathfrak{M}}_i}{{\mathfrak{M}}_{i+1}}}$ has no non-zero proper submodule.

A ring R is termed local if it possesses a single maximal ideal N and the quotient ${\displaystyle \frac{R}{N}}\cong {\mathbb{F}}_{p^m}.$ By Nakayama’s Lemma [13], when R is a finite local ring, it has a sequence of ideals represented as

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

where $t\ge 1$ is known as the nilpotency index of N, and it satisfies $t\le l_R\left(R\right)=l$. If $\mathfrak{M}$ exhibits a composition series over R, then

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

For a finite local ring R, there are integers l and m such that the ideal N has an order given by

$$\left|N\right|=p^{(l-1)m}\mathrm{if}{N}^l=0,$$

while the size of the ring R is described by

$$\left|R\right|=p^{lm}.$$

Within R, the characteristic is $p^n$, where $1\le n\le l$. Additionally, R features a prime subring $\mathcal{O}$, which forms a Galois ring identified as $GR(p^n,m)$. If ${\vartheta }_i\in N$ for $1\le i\le k,$ then (an $\mathcal{O}$-module)

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

(3)

The elements $\{{\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_k\}$ are recognized as a distinguished basis of the ring R over $\mathcal{O}$ [14]. These elements play a crucial role in the classification of rings up to isomorphism based on $p, n, m, l,$ and k. Specifically, if $k=0$, $R=\mathcal{O}=GR(p^n,m)$. When $t=1$, the ideal N becomes trivial, indicating that R is a field, which also implies $l=1$. Furthermore, in the case where $k=1$, the rings are identified as singleton local rings, extensively discussed in [15]. Therefore, our focus shifts to the condition where $t>2$.

A noteworthy type of local rings occurs when $n=l$. Based on their orders, these local rings correspond to Galois rings $GR(p^n,m)$, which are distinctly defined by $p, n,$ and m. These rings originate from ${\mathbb{Z}}_{p^n}$ and an element $f\in R\left[x\right]$ is classified as a basic irreducible polynomial when its reduction $\overline{f}$ is irreducible over ${\mathbb{F}}_{p^m}$. If we denote $g\in R\left[x\right]$ as a basic irreducible polynomial, and take $m=\mathrm{deg}\left(\overline{g}\right)$, we can find a unit $\beta$ in $R\left[x\right]$ and a monic polynomial $f\in R\left[x\right]$ such that

$$f=\beta g.$$

Letting $N=⟨p⟩$, we find that $n=l$, allowing us to express R as

$$R\cong {\mathbb{Z}}_{p^n}\left[\zeta \right]\cong {\displaystyle \frac{{\mathbb{Z}}_{p^n}\left[x\right]}{⟨g\left(x\right)⟩}},$$

where $\zeta$ denotes a root of the basic irreducible polynomial $g\left(x\right)\in {\mathbb{Z}}_{p^n}\left[x\right]$ with degree m. This essentially represents a Galois extension of ${\mathbb{Z}}_{p^n}$.

A local ring R qualifies as Frobenius when the annihilator of its Jacobson radical, ${\mathrm{Ann}}_R\left(N\right)$, serves as its sole minimum ideal. This requirement indicates that ${\mathrm{Ann}}_R\left(N\right)$ constitutes a simple ideal of R (refer to [10]). Alternatively, one can define a Frobenius ring as the summation of all minimal R-submodules within a given R-module $\mathfrak{M}$. As stated in [16], a ring R qualifies as Frobenius if the quotient ${\displaystyle \frac{R}{N}}$ exhibits an isomorphism with its socle, a condition that is satisfied when $\mathrm{soc}\left(R\right)$ is cyclic. For finite Frobenius local rings, ${\mathrm{Ann}}_R\left(N\right)=N^{t-1}$ [9]. This may not be true for more general rings, as this document will clarify. The category of Frobenius rings encompasses both chain and non-chain rings such as ${\displaystyle \frac{{\mathbb{Z}}_p[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2,{\vartheta }_2^2⟩}}\mathrm{and}{\displaystyle \frac{{\mathbb{Z}}_{p^n}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2-p^{n-1},p{\vartheta }_1⟩}},\mathrm{where}n\ge 3.$ For additional examples, we suggest referring to [9], and to [17,18,19] for more details on the topic of finite local rings.

3. Frobenius Local Rings of Length 5 with t = 4

In this section, we intend to explore all potential structures of R and list them by varying the values of n and k. Suppose $\mathfrak{M}$ is a R-module. A subset S of $\mathfrak{M}$ is considered to span $\mathfrak{M}$ if the image $\overline{S}$ in the quotient ${\displaystyle \frac{\mathfrak{M}}{N\mathfrak{M}}}$ generates ${\displaystyle \frac{\mathfrak{M}}{N\mathfrak{M}}}$. A minimal generating set for $\mathfrak{M}$ over R consists of generators obtained by lifting a basis of the ${\displaystyle \frac{R}{N}}$-vector space ${\displaystyle \frac{\mathfrak{M}}{N\mathfrak{M}}}$. We denote the count of such generators as ${\nu }_R\left(\mathfrak{M}\right)$.

Importantly, we have the following relationship:

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

(4)

Specifically, this leads to

$${\nu }_i=\nu \left(N^i\right)=l\left({\displaystyle \frac{N^i}{N^{i+1}}}\right).$$

(5)

It is worth mentioning that Ann$\left(N\right)$ is not necessarily $N^{t-1}.$ In general, we have $N^{t-1}\subseteq$ soc$\left(R\right).$ However, this is true when R is a Frobenius ring. If soc$\left(R\right)$ is principal, then so is $N^{t-1}$ because $N^{t-1}\subseteq$ soc$\left(R\right).$ Furthermore, soc$\left(R\right)\cong {\displaystyle \frac{R}{N}}$ and $N^{t-1}\cong {\displaystyle \frac{R}{N}}.$ Thus, we have $N^{t-1}=$ soc$\left(R\right).$ Since $t=4,$ we have soc$\left(R\right)=N^3,$ and

$$R\supset N\supset N^2\supset N^3\supset N^4=0.$$

(6)

In addition, $1\le n\le 4,$ and consequently $1\le k\le 4=l-1.$ Notice that ${\nu }_1\le k+1,$ and ${\nu }_2\le {\nu }_1^2.$ In general, we have

$$\begin{array}{ccc}\hfill & {\nu }_1+{\nu }_2+{\nu }_3=k,\hfill & \hfill \mathrm{if}n=1,\\ \hfill & n(k+1)\ge {\nu }_1+{\nu }_2+{\nu }_3\ge k+1,\hfill & \hfill \mathrm{if}n>1.\end{array}$$

(7)

The primary outcomes of this section are presented below. Based on $\mid R\mid =p^{5m}$ and $t=4$, $p^n\in \{p,p^2,p^3,p^4\}.$ Consequently, we categorize into subsections based on $n,$

(i)

If $n=1,$ then $\mathcal{O}={\mathbb{F}}_{p^m}.$

(ii)

$n=2,$ then $\mathcal{O}=GR(p^2,m).$

(iii)

$n=3,$ then $\mathcal{O}=GR(p^3,m).$

(iv)

$n=4,$ then $\mathcal{O}=GR(p^4,m).$

Finally, we introduce

$$\begin{array}{cc}\hfill & \Gamma \left(m\right)=⟨\zeta ⟩\cup \left\{0\right\}=\{0,1,\zeta ,{\zeta }^2,\dots ,{\zeta }^{p^m-2}\};\hfill \\ \hfill & A_1=\{{\zeta }^{2i+1}:1\le i\le {\displaystyle \frac{p^m-1}{2}}-1\}=\{\zeta \in {\Gamma }^{*}\left(m\right):\zeta \notin {\left({\Gamma }^{*}\left(m\right)\right)}^2\};\hfill \\ \hfill & A_2=\{{\zeta }^{2i}:1\le i\le {\displaystyle \frac{p^m-1}{2}}-1\}=\{\zeta \in {\Gamma }^{*}\left(m\right):\zeta \in {\left({\Gamma }^{*}\left(m\right)\right)}^2\}.\hfill \end{array}$$

(8)

3.1. Frobenius Local Rings with $n=1$

We denote R as a Frobenius local ring with characteristic $p,$ i.e., $n=1.$ The primary subring of R is given by $\mathcal{O}={\mathbb{F}}_{p^m}.$ The order of R indicates that k must equal $4.$ Therefore,

$$\begin{array}{cc}\hfill & R=\mathcal{O}+{\vartheta }_1\mathcal{O}+{\vartheta }_2\mathcal{O}+w\mathcal{O}+z\mathcal{O},\hfill \\ \hfill & N={\vartheta }_1\mathcal{O}+{\vartheta }_2\mathcal{O}+w\mathcal{O}+z\mathcal{O}.\hfill \end{array}$$

(9)

In light of Equation (7), we have ${\nu }_1+{\nu }_2+{\nu }_3=4.$ We also have ${\nu }_2\le {\nu }_1^2,$ so the only solution is ${\nu }_1=2$ and ${\nu }_2={\nu }_3=1.$ Note that ${\nu }_3=1$ is fixed during this work, as R is Frobenius.

Theorem 1.

Suppose R is a Frobenius local ring with $n=1.$ Then, R is isomorphic to a unique copy of

$$\begin{array}{cc}{\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^4,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2⟩}},& {\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^4,{\vartheta }_2^2-{\vartheta }_1^3,{\vartheta }_1{\vartheta }_2⟩}}.\end{array}$$

Proof. 

As $n=1$ and $k=4,$ ${\nu }_1=2$ and ${\nu }_2={\nu }_3=1.$ If ${\vartheta }_1,{\vartheta }_2\in N\setminus N^2,$ then ${\vartheta }_1^2,{\vartheta }_1{\vartheta }_2,{\vartheta }_2^2\in N^2.$ It must be the case that ${\vartheta }_1^2$ and ${\vartheta }_2^2$ are not both zero (otherwise, $N^3=0$), and we assume that ${\vartheta }_1^2\ne 0.$ If we allow ${\vartheta }_1{\vartheta }_2\in N^2\setminus N^3,$ then ${\vartheta }_1^2=\beta {\vartheta }_1{\vartheta }_2,$ where $\beta \in {\Gamma }^{*}\left(m\right),$ and letting ${\vartheta }_2⟼{\vartheta }_1-\beta {\vartheta }_2$ allows us to let ${\vartheta }_1{\vartheta }_2=0.$ Now, if ${\vartheta }_1^2=\beta {\vartheta }_2^2$ for some $\beta \in {\Gamma }^{*}\left(m\right),$ then we obtain ${\vartheta }_2^3={\vartheta }_1^3=0$, which leads to $N^3=0,$ which is a contradiction. Thus,

$$\begin{array}{cc}\hfill & N={\vartheta }_1\mathcal{O}+{\vartheta }_2\mathcal{O}+N^2,\hfill \\ \hfill & N^2={\vartheta }_1^2\mathcal{O}+N^3,\hfill \\ \hfill & N^3={\vartheta }_1^3\mathcal{O}.\hfill \end{array}$$

(10)

As ${\vartheta }_2^2\in N^3,$ ${\vartheta }_2^2=0$ or ${\vartheta }_2^2=\beta {\vartheta }_1^3,$ where $\beta \in {\Gamma }^{*}\left(m\right).$ If $p=3,$ then $\beta \in A_2,$ and we may therefore consider the correspondence ${\vartheta }_1⟼{\vartheta }_1$ and ${\vartheta }_2⟼\sqrt{\beta }{\vartheta }_2$ will simplify to ${\vartheta }_2^2={\vartheta }_1^3.$ In contrast, if $p\ne 3,$ then we adopt the map that sends ${\vartheta }_1⟼\sqrt[3]{\beta }{\vartheta }_1$ and ${\vartheta }_2⟼{\vartheta }_2,$ i.e., $\beta \in {\Gamma }^{*}{\left(m\right)}^3.$ In conclusion, we have the relations between ${\vartheta }_1$ and ${\vartheta }_2$:

$${\vartheta }_1^4={\vartheta }_2^2={\vartheta }_1{\vartheta }_2=0,\mathrm{or}{\vartheta }_2^2={\vartheta }_1^3,{\vartheta }_1^4={\vartheta }_1{\vartheta }_2=0.$$

Consequently, we have only 2 of such rings, namely,

$$R\cong \left\{\begin{array}{c}{\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^4,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2⟩}},\hfill \\ {\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^4,{\vartheta }_2^2-{\vartheta }_1^3,{\vartheta }_1{\vartheta }_2⟩}}.\hfill \end{array}\right.$$

 □

3.2. Frobenius Local Rings with $n=2$

This subsection handles Frobenius local rings for $n=2,$ and $\mathcal{O}=GR(p^2,m).$ We define $i_p$ to be the largest positive integer such that $p\in N^{i_p}.$ We then consider different cases: (i) $i_p=3;$ (ii) $i_p=2;$ (iii) $i_p=1.$ If $i_p=3,$ then $p\in N^3,$ and hence $pN=0$, as $N^3=$ Ann$\left(N\right).$ In such a case, we force $k=3$ by the order of R and order of $\mathcal{O}.$ The relation in (7) leads to ${\nu }_1+{\nu }_2+{\nu }_3=k+1=4,$ and thus ${\nu }_1+{\nu }_2=3.$ This will result in ${\nu }_1=2$ and ${\nu }_2=1.$ Hence,

$$\begin{array}{cc}\hfill & R=\mathcal{O}+{\vartheta }_1\mathcal{O}+{\vartheta }_2\mathcal{O}+{\vartheta }_1^2\mathcal{O},\hfill \\ \hfill & N={\vartheta }_1\mathcal{O}+{\vartheta }_2\mathcal{O}+{\vartheta }_1^2\mathcal{O}.\hfill \end{array}$$

(11)

The next theorem will treat case (i).

Theorem 2.

Suppose R is a Frobenius local ring of characteristic $p^2$ and $i_p=3.$ Then,

(i) 

If $p\ne 2,$

$$R\cong \left\{\begin{array}{c}{\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^3-p\beta ,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2,p{\vartheta }_1,p{\vartheta }_2⟩}},\hfill \\ {\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^3-p\beta ,{\vartheta }_2^2-{\vartheta }_1^2,{\vartheta }_1{\vartheta }_2,p{\vartheta }_1,p{\vartheta }_2⟩}},\hfill \\ {\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^3-p\beta ,{\vartheta }_2^2-\zeta {\vartheta }_1^2,{\vartheta }_1{\vartheta }_2,p{\vartheta }_1,p{\vartheta }_2⟩}}.\hfill \end{array}\right.$$

(ii) 

If $p=2,$

$$R\cong \left\{\begin{array}{c}{\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^3-2\beta ,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2,2{\vartheta }_1,2{\vartheta }_2⟩}},\hfill \\ {\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^3-2\beta ,{\vartheta }_2^2-{\vartheta }_1^2,{\vartheta }_1{\vartheta }_2,2{\vartheta }_1,2{\vartheta }_2⟩}},\hfill \end{array}\right.$$

where $\beta \in {\Gamma }^{*}\left(m\right).$

Proof. 

First note that $p\in N^3,$ and hence $pN=0.$ Because R is Frobenius, $N^3=⟨p⟩.$ Furthermore, the order of R and $pN=0$ force $k=3.$ Additionally, ${\nu }_1=2$ and ${\nu }_2={\nu }_3=1.$ Let ${\vartheta }_1,{\vartheta }_2$ be generators of $N\setminus N^2.$ As previously, ${\vartheta }_1^2$ and ${\vartheta }_2^2$ cannot both be 0, or we might have $N^3=0.$ One can also fix ${\vartheta }_1{\vartheta }_2=0$ by a similar argument, as in the proof of Theorem 1. Therefore, we have ${\vartheta }_2^2=\gamma {\vartheta }_1^2,$ where $\gamma \in \Gamma \left(m\right).$ As ${\vartheta }_1^3\in N^3,$ ${\vartheta }_1^3=p\beta$ for $\beta \in {\Gamma }^{*}\left(m\right).$ It is worth mentioning that ${\vartheta }_2^3=0.$ We have shown that

$$\begin{array}{cc}\hfill & N={\vartheta }_1\mathcal{O}+{\vartheta }_2\mathcal{O}+N^2,\hfill \\ \hfill & N^2={\vartheta }_1^2\mathcal{O}+N^3,\hfill \\ \hfill & N^3=p\mathcal{O}.\hfill \end{array}$$

(12)

Moreover, we have

$${\vartheta }_1^3=p\beta ,{\vartheta }_2^2=\gamma {\vartheta }_1^2,\mathrm{and}{\vartheta }_1{\vartheta }_2=0.$$

(13)

In conclusion, R is isomorphic to ${\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^3-p\beta ,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2⟩}}$ or ${\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^3-p\beta ,{\vartheta }_2^2-\gamma {\vartheta }_1^2,{\vartheta }_1{\vartheta }_2⟩}}$ depending on $\gamma =0$ or $\gamma \ne 0.$ Let $={\gamma }_i\in A_i,$ and the equations ${\vartheta }_2^2={\gamma }_1{\vartheta }_1^2$ and ${\vartheta }_2^2={\gamma }_2{\vartheta }_1^2$ define two different rings; otherwise, we obtain ${\gamma }_1\in A_2$, which is a contradiction. Thus, there are 2 forms of the last ring given that $\gamma \in A_2$ or $\gamma \in A_1.$ When $p=2,$ ${\vartheta }_2^2=\gamma {\vartheta }_1^2$ in Equation (13) will become ${\vartheta }_2^2=0$ or ${\vartheta }_2^2={\vartheta }_1^2$ because we always have $(2^m-1,2)=1,$ which means $\gamma \in A_2$ whenever $\gamma \ne 0.$ In summary, if $p=2,$ we obtain 2 of such rings when $(2^m-1,3)=1$, while we have 6 rings when $(2^m-1,3)=3$, which is the number of isomorphism classes

$$N(p=2,n=2,m,l=5,k=3)=2\times (2^m-1,3).$$

(14)

 □

Corollary 1.

The number of non-isomorphic classes of Frobenius local rings of the form ${\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^3-p\beta ,{\vartheta }_2^2-\gamma {\vartheta }_1^2,{\vartheta }_1{\vartheta }_2⟩}}$ when $p\ne 2$ is

$$\left\{\begin{array}{cc}9,\hfill & \mathit{if}{p}^m\not\equiv 1\left(\mathit{mod}3\right),\hfill \\ \\ 3,\hfill & \mathit{if}{p}^m\equiv 1\left(\mathit{mod}3\right).\hfill \end{array}\right.$$

Proof. 

The relation ${\vartheta }_2^2=\gamma {\vartheta }_1^2$ produces 3 different rings depending on $\gamma =0,\gamma \in A_1$ or $\gamma \in A_2.$ Next, we focus on ${\vartheta }_1^3=p\beta$, which have $(p^m-1,3)$ solutions [7]. Thus, the number of such rings is $3\times (p^m-1,3),$ and the results follow. □

Example 1.

If we have ${\nu }_i=1$ for all $i=1,2.$ Then, clearly we must have ${\nu }_3=2,$ and therefore,

$$\begin{array}{cc}\hfill & R=\mathcal{O}+{\vartheta }_1\mathcal{O}+{\vartheta }_1^2\mathcal{O}+{\vartheta }_1^3\mathcal{O},\hfill \\ \hfill & N={\vartheta }_1\mathcal{O}+{\vartheta }_2\mathcal{O}+N^2,\hfill \\ \hfill & N^2={\vartheta }_1^2\mathcal{O}+N^3,\hfill \\ \hfill & N^3=p\mathcal{O}+{\vartheta }_1^3\mathcal{O}.\hfill \end{array}$$

(15)

In such a case,

$$R\cong {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^4,p{\vartheta }_1⟩}}.$$

This ring is not Frobenius because soc$\left(R\right)=⟨{\vartheta }_1^3,p⟩$ is not simple.

The other cases, Case (ii) and Case (iii), will be discussed here. For (ii), $i_p=2,$ and thus $p\in N^2.$ Since $k\le 3=l-n$ and $pN\ne 0$, then by Equation (7), we have ${\nu }_1+{\nu }_2=k+1.$ Thus, we conclude that $k=2$ and ${\nu }_1=2,{\nu }_2=1,$ and

$$R=\mathcal{O}+{\vartheta }_1\mathcal{O}+{\vartheta }_2\mathcal{O}.$$

However, assuming (iii), we see that $k=2$, as in case (ii). Additionally, ${\nu }_2=2$ and ${\nu }_1=2,$ so

$$R=\mathcal{O}+{\vartheta }_1\mathcal{O}+{\vartheta }_1^2\mathcal{O}.$$

Theorem 3.

Suppose R is a Frobenius local ring with $n=2$ and $i_p\ne 3.$ Then, R is isomorphic to

(i) 

If $p\ne 2,$

$$\begin{array}{ccc}{\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-p,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2,p{\vartheta }_2⟩}},& {\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-p\zeta ,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2,p{\vartheta }_2⟩}},& R^{*}={\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-p,{\vartheta }_2^2-p{\vartheta }_1,{\vartheta }_1{\vartheta }_2,p{\vartheta }_2⟩}},\\ {\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-p\zeta ,{\vartheta }_2^2-p{\vartheta }_1,{\vartheta }_1{\vartheta }_2,p{\vartheta }_2⟩}},& {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨p{\vartheta }_1^2,{\vartheta }_1^3⟩}},& {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨p{\vartheta }_1^2,{\vartheta }_1^3-p\beta {\vartheta }_1⟩}},& \\ {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨p{\vartheta }_1^2,{\vartheta }_1^3-p{\vartheta }_1⟩}}.& \end{array}$$

(ii) 

If $p=2,$

$$\begin{array}{cccc}{\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-2,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2,p{\vartheta }_2⟩}},& {\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-2,{\vartheta }_2^2-2{\vartheta }_1,{\vartheta }_1{\vartheta }_2,2{\vartheta }_2⟩}},& {\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-2(1+\beta {\vartheta }_1),{\vartheta }_2^2-2{\vartheta }_1,{\vartheta }_1{\vartheta }_2,2{\vartheta }_2⟩}},& {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨2{\vartheta }_1^2,{\vartheta }_1^3⟩}},\\ {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨2{\vartheta }_1^2,{\vartheta }_1^3-2{\vartheta }_1⟩}}.\end{array}$$

Proof. 

First, if Case (iii) is assumed, then $p\in N\setminus N^2.$ Since $pN\ne 0,$ $p{\vartheta }_1\ne 0.$ By the order of $R,$ we have $p{\vartheta }_1^2=0.$ This implies that ${\nu }_2=2$, leading to $N^2=⟨{\vartheta }_1^2,p{\vartheta }_1⟩$ and $N^3=⟨p{\vartheta }_1⟩$, as ${\nu }_3=1.$ Therefore, we have

$$\begin{array}{cc}\hfill & N={\vartheta }_1\mathcal{O}+{\vartheta }_1^2\mathcal{O}+N^2,\hfill \\ \hfill & N^2={\vartheta }_1^2\mathcal{O}+N^3,\hfill \\ \hfill & N^3=p{\vartheta }_1\mathcal{O}.\hfill \end{array}$$

(16)

A clear relation emerges, which is ${\vartheta }_1^3=p\beta {\vartheta }_1.$ Thus, ${\vartheta }_1^3=p{\vartheta }_1\beta$ and $p{\vartheta }_1^2=0.$ One deduces easily that

$$R\cong {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^3-p\beta {\vartheta }_1,p{\vartheta }_1^2⟩}}.$$

Certainly ${\vartheta }_1^3-p\beta {\vartheta }_1={\vartheta }_1({\vartheta }_1^2-p\beta )=0.$ Thus, if $\beta \in {\Gamma }^{*}\left(m\right),$ then either $\beta \in A_1$ or $\beta \in A_2.$ Each case of $\beta$ determines a unique copy of $R.$ That is, ${\vartheta }_1⟼\sqrt{{\beta }^{-1}}{\vartheta }_1$ if $\beta \in A_2.$ Therefore, if $p\ne 2,$ there are 3 rings of the form

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^3,p{\vartheta }_1^2⟩}},\hfill \\ {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^3-p{\vartheta }_1,p{\vartheta }_1^2⟩}},\hfill \\ {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^3-p\zeta {\vartheta }_1,p{\vartheta }_1^2⟩}}.\hfill \end{array}\right.$$

In the case of $p=2,$ there are exactly 2 of such rings:

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^3,2{\vartheta }_1^2⟩}},\hfill \\ {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^3-2{\vartheta }_1,2{\vartheta }_1^2⟩}}.\hfill \end{array}\right.$$

Second, Case (ii) when $p\in N^2\setminus N^3.$ The condition $pN\ne 0$ means that $p{\vartheta }_1\ne 0$ or $p{\vartheta }_2\ne 0$ or both. The order of R shows that only one of them is satisfied, and let $p{\vartheta }_1\ne 0.$ We have $k=2,$ so we may write

$$\begin{array}{cc}\hfill & N={\vartheta }_1\mathcal{O}+{\vartheta }_2\mathcal{O}+N^2,\hfill \\ \hfill & N^2=p\mathcal{O}+N^3,\hfill \\ \hfill & N^3=p{\vartheta }_1\mathcal{O}.\hfill \end{array}$$

(17)

Hence, ${\vartheta }_1^2=p\beta +p{\beta }_1{\vartheta }_1$ and ${\vartheta }_2^2=p{\gamma }_0+p\gamma {\vartheta }_1,$ where $\beta ,{\beta }_1,\gamma ,{\gamma }_0\in \Gamma \left(m\right).$ Additionally, ${\vartheta }_1{\vartheta }_2\in N^2$, so ${\vartheta }_1{\vartheta }_2=p\delta +p{\delta }_1{\vartheta }_1,$ where $\delta ,{\delta }_1\in \Gamma \left(m\right).$ If $p\ne 2,$ then by completing squares, we obtain ${\vartheta }_1^2=p\beta .$ Note that ${\vartheta }_2^2{\vartheta }_1=p{\gamma }_0{\vartheta }_1={\vartheta }_1{\vartheta }_2^2=0$. Thus, ${\gamma }_0=0,$ and thus ${\vartheta }_2^2=p\gamma {\vartheta }_1.$ Similar discussion shows that ${\vartheta }_1^2{\vartheta }_2=p\delta {\vartheta }_1=0$; therefore, $\delta =0,$ which implies that ${\vartheta }_1{\vartheta }_2=p{\delta }_1{\vartheta }_1.$ This enables us to assume ${\vartheta }_1{\vartheta }_2=0$ by replacing ${\vartheta }_2⟼{\vartheta }_2-p{\delta }_1.$ Moreover, if $\gamma \ne 0,$ then one can check that ${\vartheta }_2^2=p{\vartheta }_1.$ Thus, our desired ring can be expressed in general as

$$R\cong {\displaystyle \frac{\mathcal{O}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-p\beta ,{\vartheta }_2^2-p\gamma {\vartheta }_1,{\vartheta }_1{\vartheta }_2,p{\vartheta }_2⟩}}.$$

As usual, one can see that, if $\gamma =0,$ there are 2 such rings where $\beta$ is in $A_1$ or in $A_2.$ If we suppose that $\gamma =1$, then, if $\beta \in A_1,$ there is a unique copy of such a ring. Meanwhile, if $\beta \in A_2,$ then there is a ${\beta }_1$ such that ${\beta }_1^2=\beta$; thus, one can check that, if ${\beta }_1\in A_2,$ we obtain a relation ${\beta }_2^4=\beta$ for some ${\beta }_2\in {\Gamma }^{*}\left(m\right).$ When $(p^m-1,4)=4,$ we have 2 copies of R in the case of $\beta \in A_2.$ However, if $(p^m-1,4)=2,$ we obtain a version R in the same case. In summary, we have 6 when $(p^m-1,4)=4$ and 4 if $(p^m-1,4)=2.$ In other words, the number of $R^{*}$ is either 1 or $3.$ Finally when $p=2,$ then ${\vartheta }_1^2=2+2\beta {\vartheta }_1$; however, since $(2^m-1,2)=1,$ there are 3 rings according to $({\vartheta }_1^2=2,{\vartheta }_2^2=0),$ $({\vartheta }_1^2=2,{\vartheta }_2^2=2{\vartheta }_1)$ or $({\vartheta }_1^2=2(1+\beta {\vartheta }_1),{\vartheta }_2^2=2{\vartheta }_1).$ □

3.3. Frobenius Local Rings with $n=3$

In this part, we classify Frobenius local rings with $n=3$ and $\mathcal{O}=GR(p^3,m).$

Theorem 4.

Suppose R is a Frobenius local ring with invariants $p,n=3,m,5,k.$ Then,

$$R\cong {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^3-p^2\beta ,p{\vartheta }_1⟩}},$$

(18)

where $\beta \in {\Gamma }^{*}\left(m\right).$

Proof. 

As $n=3$ and $t=4,$ then $p\in N\setminus N^2.$ By Equation (7) and $N^3\ne 0$, then $k=2.$ Since $k=2,$

$$R=\mathcal{O}+{\vartheta }_1\mathcal{O}+{\vartheta }_2\mathcal{O}.$$

If ${\nu }_1=3$ and $N=⟨p,{\vartheta }_1,{\vartheta }_2⟩,$ then $p{\vartheta }_1=p{\vartheta }_2=0$ because of the order of $R.$ Additionally, $N^2=⟨p^2,{\vartheta }_1^2,{\vartheta }_2^2⟩,$ and since ${\vartheta }_1^2,{\vartheta }_2^2\in \mathcal{O},$ $N^2=⟨p^2⟩.$ Note that $p^{2}N=0$ leads to $N^3=0$, which is a contradiction since $t=4.$ Hence, ${\nu }_1=2$ and ${\nu }_2=2.$ Moreover, $p{N}^2=p^{2}N=0,$ so

$$\begin{array}{cc}\hfill & R=\mathcal{O}+{\vartheta }_1\mathcal{O}+{\vartheta }_1^2\mathcal{O},\hfill \\ \hfill & N=p\mathcal{O}+{\vartheta }_1\mathcal{O}+N^2,\hfill \\ \hfill & N^2={\vartheta }_1^2\mathcal{O}+N^3,\hfill \\ \hfill & N^3={\vartheta }_1^3\mathcal{O}.\hfill \end{array}$$

(19)

Therefore, we have the relation among generators:

$${\vartheta }_1^3=p^2\beta ,\mathrm{and}p{\vartheta }_1=0,$$

where $\beta \in {\Gamma }^{*}\left(m\right).$ In this case, the structure of R is expressed as

$$R\cong {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^3-p^2\beta ,p{\vartheta }_1⟩}}.$$

There are $(3,p^m-1),$ as we saw above, of such rings. To conclude, we have 3 rings if $p^m\equiv 1(\mathrm{mod}3$) and 1 ring otherwise. □

3.4. Frobenius Local Rings with $n=4$

Observe that $k=1$ and $\mathcal{O}=GR(p^4,m)$. Therefore,

$$R=\mathcal{O}+{\vartheta }_1\mathcal{O}.$$

Before we delve into classification of Frobenius local rings, we present the following results to characterize Frobenius local rings based just on their invariants $p,n=4,m,l,k=1.$ We first assume $t_{{\vartheta }_1}$ to be the least positive integer such that $p^{t_{{\vartheta }_1}}{\vartheta }_1=0.$

Lemma 1.

Let $k=1.$ Then,

$$t=\left\{\begin{array}{cc}n+i_p-1,\hfill & \mathit{if}{t}_{{\vartheta }_1}\le n-1,\hfill \\ n+i_p,\hfill & \mathit{if}{t}_{{\vartheta }_1}=n.\hfill \end{array}\right.$$

Proof. 

Assume that $k=1$. Then we have $N=⟨p,{\vartheta }_1⟩$. We will analyze two scenarios: one where ${\nu }_1=1$ and another where ${\nu }_1=2$. In the first case, when $\nu \left(N\right)=1$, the ring R is a chain, which implies that $t=n+i_p$. In the second case, for a non-chain scenario, we find that $p\in N\setminus N^2$, leading to $i_p=1$. Therefore, we can express $N^i$ as $⟨p^i,p^{i-1}{\vartheta }_1,{\vartheta }_1^i⟩$. Since ${\vartheta }_1^i\in ⟨p^i,p^{i-1}{\vartheta }_1⟩$ and given that $i\ge 2$ (because $k=1$), we can deduce that $N^i=⟨p^i,p^{i-1}{\vartheta }_1⟩$. Furthermore, since $t\ge n$ and $N^n=⟨p^n,p^{n-1}{\vartheta }_1⟩$, we can conclude that $t=n$ if $t_{{\vartheta }_1}\le n-1$, and $t=n+1$ when $t_{{\vartheta }_1}=n$. □

If $k=1,$ then $R=\mathcal{O}+{\vartheta }_1\mathcal{O}.$ Additionally, we have

$${\vartheta }_1^2=p^d\beta h+p^{e}u{\beta }_1{h}_1,$$

(20)

where $\beta ,{\beta }_1\in {\Gamma }^{*}\left(m\right)$ and $h,h_1\in H.$ Note that ${\vartheta }_1^2=0$ when $d=n$ and $e=t_{{\vartheta }_1}.$

Proposition 1.

If R is a local non-chain ring with $k=1$, then R is Frobenius if and only if ${\vartheta }_1\notin$ Ann$\left(N\right)$ and $t_{{\vartheta }_1}\ne n-1.$

Proof. 

First, let us assume that R is a Frobenius ring; thus, $soc\left(R\right)$ is both simple and unique. Furthermore, since $soc\left(R\right)$ is principal, we can denote it as $soc\left(R\right)=⟨w⟩$. This implies that $pw=w{\vartheta }_1=0$. Given that $w=r_0+r_1{\vartheta }_1$, we can observe that $r_0\in p^{n-1}GR(p^n,m)$ and $r_1\in p^{t_{{\vartheta }_1}-1}GR(p^n,m)$. Consequently, we have

$$w=p^{n-1}{v}_0+p^{t_{{\vartheta }_1}-1}{v}_1{\vartheta }_1,$$

(21)

where $v_0,v_1\in \Gamma \left(m\right)$ with at least one of them being non-zero. Therefore, $w\in ⟨p^{n-1},p^{t_{{\vartheta }_1}-1}{\vartheta }_1⟩$. Since $t-1\le n-1$, it follows that $soc\left(R\right)\subseteq ⟨p^{t_{{\vartheta }_1}-1}⟩$. Hence, we conclude that $soc\left(R\right)=p^{n-1}R$. Now, we can examine two cases. In the first case, if $v_0=0$, then $soc\left(R\right)=⟨p^{t_{{\vartheta }_1}-1}{\vartheta }_1⟩$, which means $p^{n-1}{\vartheta }_1$ lies in the annihilator of N. Consequently, $p^{n-1}{\vartheta }_1\in soc\left(R\right)$, implying $n=t_{{\vartheta }_1}$. In the second case, let us assume $v_0\ne 0$. Since $wu=0$ from Equation (21), it follows that $p^{n-1}{\vartheta }_1=0$ and $p^{t_{{\vartheta }_1}-1}{\vartheta }_1^2=0$, leading to $p^{d+t_{{\vartheta }_1}-1}\beta h=0$. Thus, we infer that $t_{{\vartheta }_1}\ne n-1$ and ${\vartheta }_1\notin \mathrm{Ann}\left(N\right)$. Conversely, if we assume $t_{{\vartheta }_1}\ne n-1$, then $p^{n-1}{\vartheta }_1\ne 0$ and $p^{n-1}{\vartheta }_{1}N=0$. Hence, $soc\left(R\right)=⟨p^{n-1}{\vartheta }_1⟩$ when $t_{{\vartheta }_1}=n$, and $soc\left(R\right)=⟨p^{n-1}⟩$ when $t_{{\vartheta }_1}<n-1$. This indicates that R is indeed Frobenius. □

Based on Proposition 1, we obtain

$$\mathrm{soc}\left(R\right)=\left\{\begin{array}{cc}⟨p^{n-1}⟩,\hfill & \mathrm{if}{t}_{{\vartheta }_1}<n-1,\hfill \\ ⟨p^{n-1}{\vartheta }_1⟩,\hfill & \mathrm{if}{t}_{{\vartheta }_1}=n.\hfill \end{array}\right.$$

This example is quite helpful, as it demonstrates that Ann$\left(R\right)$ is not necessarily $N^{t-1}$, contrary to the assertion made in [5].

Example 2.

Let $R={\displaystyle \frac{{\mathbb{Z}}_{p^3}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2,p{\vartheta }_1⟩}}.$ Then, $t_{{\vartheta }_1}=1,$ and $t=3.$ This implies, by Theorem 1, that R is not Frobenius. This is true because $N^2=⟨p^2⟩$ while soc$\left(R\right)=⟨p^2,{\vartheta }_1⟩.$ This means that $N^2\subset$ soc$\left(R\right).$ This example shows also that the condition ${\vartheta }_1\notin$ Ann$\left(N\right)$ is necessary.

Theorem 5.

Suppose R is a Frobenius local ring and $n=4.$ Then, R is isomorphic to a unique copy of

(i) 

If $p\ne 2,$

$$\begin{array}{ccc}{\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2,p{\vartheta }_1⟩}},& {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2-p^3,p{\vartheta }_1⟩}},& {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2-p^3\zeta ,p{\vartheta }_1⟩}}.\end{array}$$

(ii) 

If $p=2,$

$$\begin{array}{cc}{\displaystyle \frac{GR(16,m)\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2,2{\vartheta }_1⟩}},& {\displaystyle \frac{GR(16,m)\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2-8,2{\vartheta }_1⟩}}.\end{array}$$

Proof. 

We have $k=1$ by Equation (7). The order of R explains that $p{\vartheta }_1=0$ and $p{N}^2\ne 0$. Hence, ${\vartheta }_1^2\notin N^2\setminus N^3$; otherwise, it gives ${\vartheta }_1^2=p^2\beta .$ When $\beta \ne 0,$ we obtain $p{\vartheta }_1^2=p^3\beta \ne 0,$ and this is not possible since $\mid R\mid =p^{5m}.$ This implies that ${\vartheta }_1^2\in N^3,$ and thus ${\nu }_1=2$ and ${\nu }_2=1,$ so

$$\begin{array}{cc}\hfill & R=\mathcal{O}+{\vartheta }_1\mathcal{O},\hfill \\ \hfill & N=p\mathcal{O}+{\vartheta }_1\mathcal{O}+N^2,\hfill \\ \hfill & N^2=p^{2}GR(p^3,m)+N^3,\hfill \\ \hfill & N^3=p^{3}GR(p^3,m).\hfill \end{array}$$

(22)

Moreover,

$${\vartheta }_1^2=p^3\beta ,p{\vartheta }_1=0,$$

where $\beta \in \Gamma \left(m\right).$ The construction of R is then

$$R\cong {\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2-p^3\beta ,p{\vartheta }_1⟩}}.$$

When $p\ne 2,$ there are 3 associated with $\beta =0,\beta \in A_2$ or $\beta \in A_1$, while if $p=2,$ there are 2 copies of such rings. □

This completes the classification of all Frobenius local rings of length 5 and of order $p^{5m}.$ (Table 1)

Table 1. Classes of local non-chain rings of length 5 and of order $p^{5m}$ with $t=4$.

char$\mathbf{\left(}\mathit{R}\mathbf{\right)}\mathbf{=}{\mathit{p}}^{\mathit{n}}$	Non-Isomorphic Classes
	$\mathit{p}\mathbf{\ne }\mathbf{2}$	$\mathit{p}\mathbf{=}\mathbf{2}$
p	${\displaystyle \frac{{\mathbb{F}}_{p^m}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^4,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2⟩}}$	${\displaystyle \frac{{\mathbb{F}}_{2^m}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^4,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2⟩}}$
	${\displaystyle \frac{{\mathbb{F}}_{p^m}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^4,{\vartheta }_2^2-{\vartheta }_1^3,{\vartheta }_1{\vartheta }_2⟩}}$	${\displaystyle \frac{{\mathbb{F}}_{2^m}[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^4,{\vartheta }_2^2-{\vartheta }_1^3,{\vartheta }_1{\vartheta }_2⟩}}$
	${\displaystyle \frac{GR(p^2,m)\left[{\vartheta }_1\right]}{⟨p{\vartheta }_1^2,{\vartheta }_1^3⟩}}$	${\displaystyle \frac{GR(4,m)\left[{\vartheta }_1\right]}{⟨2{\vartheta }_1^2,{\vartheta }_1^3⟩}}$
$p^2$ and $i_p=1$	${\displaystyle \frac{GR(p^2,m)\left[{\vartheta }_1\right]}{⟨p{\vartheta }_1^2,{\vartheta }_1^3-p\beta {\vartheta }_1⟩}}$	${\displaystyle \frac{GR(4,m)\left[{\vartheta }_1\right]}{⟨2{\vartheta }_1^2,{\vartheta }_1^3-2{\vartheta }_1⟩}}$
	${\displaystyle \frac{GR(p^2,m)\left[{\vartheta }_1\right]}{⟨p{\vartheta }_1^2,{\vartheta }_1^3-p{\vartheta }_1⟩}}$
	${\displaystyle \frac{GR(p^2,m)[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-p,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2,p{\vartheta }_2⟩}}$	${\displaystyle \frac{GR(4,m)[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-2,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2,p{\vartheta }_2⟩}}$
$p^2$ and $i_p=2$	${\displaystyle \frac{GR(p^2,m)[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-p\zeta ,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2,p{\vartheta }_2⟩}}$	${\displaystyle \frac{GR(4,m)[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-2,{\vartheta }_2^2-2{\vartheta }_1,{\vartheta }_1{\vartheta }_2,2{\vartheta }_2⟩}}$
	${\displaystyle \frac{GR(p^2,m)[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-p,{\vartheta }_2^2-p{\vartheta }_1,{\vartheta }_1{\vartheta }_2,p{\vartheta }_2⟩}}$	${\displaystyle \frac{GR(4,m)[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-2(1+\beta {\vartheta }_1),{\vartheta }_2^2-2{\vartheta }_1,{\vartheta }_1{\vartheta }_2,2{\vartheta }_2⟩}}$
	${\displaystyle \frac{GR(p^2,m)[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^2-p\zeta ,{\vartheta }_2^2-p{\vartheta }_1,{\vartheta }_1{\vartheta }_2,p{\vartheta }_2⟩}}$
	${\displaystyle \frac{GR(p^2,m)[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^3-p\beta ,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2,p{\vartheta }_1,p{\vartheta }_2⟩}}$	${\displaystyle \frac{GR(4,m)[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^3-2\beta ,{\vartheta }_2^2-{\vartheta }_1^2,{\vartheta }_1{\vartheta }_2,2{\vartheta }_1,2{\vartheta }_2⟩}}$
$p^2$ and $i_p=3$	${\displaystyle \frac{GR(p^2,m)[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^3-p\beta ,{\vartheta }_2^2-{\vartheta }_1^2,{\vartheta }_1{\vartheta }_2,p{\vartheta }_1,p{\vartheta }_2⟩}}$	${\displaystyle \frac{GR(4,m)[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^3-2\beta ,{\vartheta }_2^2,{\vartheta }_1{\vartheta }_2,2{\vartheta }_1,2{\vartheta }_2⟩}}$
	${\displaystyle \frac{GR(p^2,m)[{\vartheta }_1,{\vartheta }_2]}{⟨{\vartheta }_1^3-p\beta ,{\vartheta }_2^2-\zeta {\vartheta }_1^2,{\vartheta }_1{\vartheta }_2,p{\vartheta }_1,p{\vartheta }_2⟩}}$
$p^3$	${\displaystyle \frac{GR(p^3,m)\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^3-p^2\beta ,p{\vartheta }_1⟩}}$	${\displaystyle \frac{GR(8,m)\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^3-p^2\beta ,p{\vartheta }_1⟩}}$
	${\displaystyle \frac{GR(p^4,m)\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2,p{\vartheta }_1⟩}}$	${\displaystyle \frac{GR(16,m)\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2,2{\vartheta }_1⟩}}$
$p^4$	${\displaystyle \frac{GR(p^4,m)\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2-p^3,p{\vartheta }_1⟩}}$	${\displaystyle \frac{GR(16,m)\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2-8,2{\vartheta }_1⟩}}$
	${\displaystyle \frac{GR(p^4,m)\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2-p^3\zeta ,p{\vartheta }_1⟩}}$

Example 3.

If $m=2,$ we enumerate all Frobenius local rings of length $5$ and $t=4.$

This example demonstrates how these rings can be used in the construction of linear codes. In general, R-submodules of $R^s$ are all linear codes of length s over $R.$ Specifically, ideals of the quotient ring ${\displaystyle \frac{R\left[x\right]}{⟨x^s-1⟩}}$ are cyclic codes of length s over R. Further details about the history of linear codes over finite rings may be found in [5,9,20].

Example 4.

Suppose $R={\displaystyle \frac{\mathcal{O}\left[{\vartheta }_1\right]}{⟨{\vartheta }_1^2-8,2{\vartheta }_1⟩}},$ where $\mathcal{O}={\mathbb{Z}}_{16}.$ Then, by Theorem 5, we have that R is a Frobenius local ring of order $p^5$, and $\{{\vartheta }_1,2\}$ is a minimal generating set for N, the maximal ideal of R, $N^3=⟨8⟩$. Moreover, we have $\Gamma \left(1\right)=\{0,1\}\subset \mathcal{O}$. As R is Frobenius local, all of its ideals exist in the chain:

$$⟨0⟩\subset N^3=⟨8⟩\subset N^2=⟨4⟩\subset N=⟨{\vartheta }_1,2⟩\subset R.$$

The chains of proper ideals of R are of the form

$$\begin{array}{cc}\hfill & ⟨0⟩\subset ⟨8⟩\subset ⟨{\vartheta }_1⟩\subset N,\hfill \\ \hfill & ⟨0⟩\subset ⟨8⟩\subset ⟨4⟩\subset ⟨2⟩\subset N,\hfill \\ \hfill & ⟨0⟩\subset ⟨8⟩\subset ⟨\alpha {\vartheta }_1+4⟩\subset ⟨{\vartheta }_1,4⟩\subset N,(\ast )\hfill \\ \hfill & ⟨0⟩\subset ⟨8⟩\subset ⟨4⟩\subset ⟨\alpha {\vartheta }_1+2⟩\subset N,\hfill \end{array}$$

where $\alpha \in \Gamma \left(1\right).$ Suppose also that $s=7$. Then $x^7-1=(x+1)(x^3+x+1)(x^3+x^2+1)$ over ${\mathbb{F}}_2,$ so 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 (as a direct sum),

$${\displaystyle \frac{R\left[x\right]}{⟨x^7-1⟩}}\cong {\displaystyle \frac{R\left[x\right]}{⟨q_1\left(x\right)⟩}}+{\displaystyle \frac{R\left[x\right]}{⟨q_2\left(x\right)⟩}}+{\displaystyle \frac{R\left[x\right]}{⟨q_3\left(x\right)⟩}},$$

by the Chinese Remainder Theorem. Let us denote $R_1=R={\displaystyle \frac{R\left[x\right]}{⟨q_1\left(x\right)⟩}},$ $R_2={\displaystyle \frac{R\left[x\right]}{⟨q_2\left(x\right)⟩}}$ and $R_3={\displaystyle \frac{R\left[x\right]}{⟨q_3\left(x\right)⟩}}.$ Furthermore, we let $\Gamma \left(3\right)=\{{\alpha }_0+{\alpha }_{1}x+{\alpha }_2{x}^2:{\alpha }_i\in \Gamma \left(1\right)\}$ for the rings $R_2$ and $R_3.$ The ideal lattices of $R_2$ and $R_3$ are similar to that of R in (*) while taking $\alpha \in \Gamma \left(3\right).$ Thus, if C is a cyclic code over R of length $7,$ then C is an ideal of ${\displaystyle \frac{R\left[x\right]}{⟨x^7-1⟩}},$ and can be expressed by the form (as a direct sum)

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

where $C_1,C_2$, and $C_3$ are cyclic codes over $R,$ $R_1$, and $R_3,$ respectively. Based on relation in (*), we describe some of algebraic structures of cyclic codes over $R,$

(i) 

If $C_1=⟨{\vartheta }_1⟩,$ $C_2=⟨{\beta }_1{\vartheta }_1+4⟩$ and $C_3=⟨{\beta }_2{\vartheta }_1+2⟩,$ where ${\beta }_i$ are in $\Gamma \left(3\right).$ This means that the cyclic codes, in $R+R_2+R_3$, are of the form

$$C\cong ⟨{\vartheta }_1⟩+⟨{\beta }_1{\vartheta }_1+4⟩+⟨{\beta }_2{\vartheta }_1+2⟩,$$

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

$$C=⟨{\vartheta }_1{q}_2\left(x\right)q_3\left(x\right),({\beta }_1{\vartheta }_1+4)q_1\left(x\right)q_3\left(x\right),({\beta }_2{\vartheta }_1+2)q_1\left(x\right)q_2\left(x\right)⟩.$$

(ii) 

If $C_1=⟨\alpha {\vartheta }_1+4⟩,$ $C_2=⟨{\theta }_1,4⟩$ and $C_3=⟨{\beta }_1{\vartheta }_1+2⟩,$ where $\alpha \in \Gamma \left(1\right)$ and ${\beta }_1\in \Gamma \left(3\right).$ Then, in $R_1+R_2+R_3$, we have

$$C\cong ⟨\alpha {\vartheta }_1+4⟩+⟨{\vartheta }_1,4⟩+⟨{\beta }_1{\vartheta }_1+2⟩.$$

In ${\displaystyle \frac{R\left[x\right]}{⟨x^7-1⟩}}$, this corresponds to

$$C=⟨(\alpha {\vartheta }_1+4)q_2\left(x\right)q_3\left(x\right),{\vartheta }_1{q}_1\left(x\right)q_3\left(x\right),4q_1\left(x\right)q_3\left(x\right),({\beta }_1{\vartheta }_1+2)q_1\left(x\right)q_2\left(x\right)⟩.$$

(iii) 

Assume that $C_1=⟨2⟩,$ $C_2=⟨{\theta }_1+4⟩$ and $C_3=⟨{\beta }_1{\vartheta }_1+4⟩,$ where ${\beta }_1\in \Gamma \left(3\right).$ In $R_1+R_2+R_3$. We obtain

$$C\cong ⟨2⟩+⟨{\vartheta }_1+4⟩+⟨{\beta }_1{\vartheta }_1+4⟩.$$

Then, as a correspondent in the ring ${\displaystyle \frac{R\left[x\right]}{⟨x^7-1⟩}}$, the code C is of the form

$$C=⟨2q_2\left(x\right)q_3\left(x\right),({\vartheta }_1+4)q_1\left(x\right)q_3\left(x\right),({\beta }_1{\vartheta }_1+2)q_1\left(x\right)q_2\left(x\right)⟩.$$

4. Conclusions

The significance of local rings defined by the parameters $p, n, m, l$, and k in coding theory has made them a crucial area of research. In particular, Frobenius local rings have garnered attention due to their relevance in coding theory. In this paper, we explored local rings of length 5, specifically those with $l=5$ and $t=4.$ We successfully categorized all such rings, up to isomorphism, based on their invariants. Moreover, we provided a count of Frobenius rings that meet these criteria in Table 2. As a result, we compiled a comprehensive list of Frobenius local rings of orders $2^{10}$, $3^{10}$, and $5^{10}$ in Table 3.

Table 2. Numbers of local non-chain rings of length 5 and of order $p^{5m}$ with $t=4$.

char$\mathbf{\left(}\mathit{R}\mathbf{\right)}\mathbf{=}{\mathit{p}}^{\mathit{n}}$	Number of Non-Isomorphic Classes
	$\mathit{p}\mathbf{\ne }\mathbf{2}$	$\mathit{p}\mathbf{=}\mathbf{2}$
p	2	2
$p^2$ and $i_p=1$	3	2
$p^2$ and $i_p=2$	$\left\{\begin{array}{cc}6,\hfill & \mathrm{if}{p}^m\not\equiv 1\left(\mathrm{mod}4\right)\hfill \\ 4,\hfill & \mathrm{if}{p}^m\equiv 1\left(\mathrm{mod}4\right)\hfill \end{array}\right.$	3
$p^2$ and $i_p=3$	$\left\{\begin{array}{cc}9,\hfill & \mathrm{if}{p}^m\not\equiv 1\left(\mathrm{mod}3\right)\hfill \\ 3,\hfill & \mathrm{if}{p}^m\equiv 1\left(\mathrm{mod}3\right)\hfill \end{array}\right.$	$\left\{\begin{array}{cc}6,\hfill & \mathrm{if}{2}^m\not\equiv 1\left(\mathrm{mod}3\right)\hfill \\ 2,\hfill & \mathrm{if}{2}^m\equiv 1\left(\mathrm{mod}3\right)\hfill \end{array}\right.$
$p^3$	$\left\{\begin{array}{cc}3,\hfill & \mathrm{if}{p}^m\not\equiv 1\left(\mathrm{mod}3\right)\hfill \\ 1,\hfill & \mathrm{if}{p}^m\equiv 1\left(\mathrm{mod}3\right)\hfill \end{array}\right.$	$\left\{\begin{array}{cc}3,\hfill & \mathrm{if}{2}^m\not\equiv 1\left(\mathrm{mod}3\right)\hfill \\ 1,\hfill & \mathrm{if}{2}^m\equiv 1\left(\mathrm{mod}3\right)\hfill \end{array}\right.$
$p^4$	3	2

Table 3. Numbers of chain rings of length 5 and of order $p^{10}$ with $t=4$.

char$\mathbf{\left(}\mathit{R}\mathbf{\right)}\mathbf{=}{\mathit{p}}^{\mathit{n}}$	Number of Non-Isomorphic Classes
	$\mathit{p}\mathbf{=}\mathbf{5}$	$\mathit{p}\mathbf{=}\mathbf{3}$	$\mathit{p}\mathbf{=}\mathbf{2}$
p	2	2	2
$p^2$ and $i_p=1$	3	3	2
$p^2$ and $i_p=2$	4	4	3
$p^2$ and $i_p=3$	3	9	2
$p^3$	1	3	1
$p^4$	3	3	2