Frobenius Local Rings of Order p^4m

Abstract

Suppose R is a finite commutative local ring, then it is known that R has four positive integers $p,n,m,k$ called the invariants of R, where p is a prime number. This paper investigates the structure and classification up to isomorphism of local rings with residue field ${\mathbb{F}}_{p^m}$ and of length 4. Specifically, it gives a comprehensive characterization of Frobenius local rings of order $p^{4m}$. Furthermore, we provide a detailed enumeration of the classes of all such rings with respect to their invariants $p,n,m,k$. Finite Frobenius rings are particularly advantageous for coding theory. This suitability arises from the fact that two classical theorems by MacWilliams, the Extension Theorem and the MacWilliams relations for symmetrized weight enumerators, can be generalized from finite fields to finite Frobenius rings.

1. Introduction

This article assumes that all rings have an identity element, are finite, and are commutative with a residue field of order $p^m$, it is denoted by ${\mathbb{F}}_{p^m}$, where p is a prime number. Furthermore, it is supposed that all modules over these rings are considered to be finitely generated. A ring R is said to be local if it has a unique maximal ideal, represented by J, called the radical of R. In the event that each ideal within R is generated with one element, R is a principal ideal ring (PIR). A very special class of PIRs appears when J is principal; this is called the class of chain rings [1,2]. Let M be a module over R. The total sum of all minimal R-submodules of M is defined as the socle of M, or $\mathrm{soc}\left(M\right)$. A ring R is said to be Frobenius if $\mathrm{soc}\left(R\right)\cong {\displaystyle \frac{R}{J}}={\mathbb{F}}_{p^m}$ when R is considered as an R-module. In this paper, we single out Frobenius rings because of their significant role in coding theory. The fact that Frobenius rings meet both MacWilliams theorems is one of the key reasons they are thought to be the proper class to characterize codes. Moreover, we can relate the symmetrized weight enumerators of codes and their duals, and determine the generating characters by breaking down Frobenius local rings into their primary components. For more information on the characterization of Frobenius rings, we recommend [3,4], and for symmetrized weight enumerators of linear codes over these rings, see [5,6,7].

Local rings of length $l=1$ and $l=2$ have their J principal, and thus they are chain rings. There is only one local ring with $l=1$ elements: the Galois field ${\mathbb{F}}_p$. While, the local rings of length $l=2$, i.e., of order $p^{2m}$ are: $GR(p^2,m)$ and ${\displaystyle \frac{{\mathbb{F}}_{p^m}\left[u\right]}{\langle u^2\rangle }}$. When $l=3$, the classes of local rings of order $p^{3m}$ are: If p is odd, the Frobenius local rings are: $GR(p^3,m)$, ${\displaystyle \frac{{\mathbb{F}}_{p^m}\left[u\right]}{\langle u^3\rangle }}$, ${\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^2-p,pu\rangle }}$ and ${\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^2-p\beta ,pu\rangle }}$, where $\alpha$ is a generator of the multiplication group ${\mathbb{F}}_{p^m}^{*};$ If $p=2$, the Fobenius local rings are: $GR(8,m)$, ${\displaystyle \frac{{\mathbb{F}}_{2^m}\left[u\right]}{\langle u^3\rangle }}$ and ${\displaystyle \frac{GR(4,m)\left[u\right]}{\langle u^2-2,2u\rangle }}$ (see [2]). The rings ${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{{\langle u,v\rangle }^2}}$, ${\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^2,pu\rangle }}$, ${\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{{\langle u,v\rangle }^2}}$ and ${\displaystyle \frac{GR(8,m)\left[u\right]}{\langle u^2,2u\rangle }}$ are not Frobenius local rings, because the annihilator of their maximal ideals soc$\left(R\right)$ is not a simple ideal, and the other rings are chain rings.

If A is an R-module over a finite local ring R with residue field ${\mathbb{F}}_{p^m}$ and l is the length of A, the relation $\mid A\mid ={\mid {\mathbb{F}}_{p^m}\mid }^l$, explained later in the Preliminary section, implies that if R is a Frobenius local ring with $p^{4m}$ elements, then R has length $l=4$. However, we show later that there are many finite Frobenius local non-chain rings of length 4 for various values of the nilpotency index t, of J, where $1\le t\le 4$. A particular class of such rings was considered in the literature when examining constacyclic codes [8], where $m=1$ and $t=3$. The ring ${\displaystyle \frac{{\mathbb{Z}}_{p^4}\left[u\right]}{\langle u^2-p^3\beta ,pu\rangle }}$, with $p^4$ elements was introduced in [7], where linear codes over this alphabet have been examined. It was recently demonstrated in [6] that when linear codes over a specific class of these rings were determined, the significance of Frobenius local rings of order $p^{4m}$ was established. More recently, authors of [2] studied linear codes over local rings of order $p^4$. This paper aims to accomplish two goals. The structure of finite Frobenius local rings of length four $l=4$ (i.e., with nilpotency index less than or equal four) must first be concluded. Secondly, the classification up to isomorphism of finite Frobenius rings is determined and the classes are enumerated with respect to their invarinats $p,n,m,k$. All Frobenius local non-chain and chain rings with $p^8$ elements are subsequently listed in

Table 1. Frobenius local rings of orders $p^8$, 16 and 256.

Rings of Order ${\mathit{p}}^8$, $\mathit{p}\ne 2$	Rings of Order 16	Rings of Order 256
${\displaystyle \frac{{\mathbb{F}}_{p^2}[u,v]}{\langle u^2-v^2,uv\rangle }}$	${\displaystyle \frac{{\mathbb{F}}_2[u,v]}{\langle u^2-v^2,uv\rangle }}$	${\displaystyle \frac{{\mathbb{F}}_4[u,v]}{\langle u^2-v^2,uv\rangle }}$
${\displaystyle \frac{{\mathbb{F}}_{p^2}[u,v]}{\langle u^2-\alpha v^2,uv\rangle }}$	${\displaystyle \frac{GR(4,1)\left[u\right]}{\langle u^2-2\rangle }}$	${\displaystyle \frac{GR(4,2)\left[u\right]}{\langle u^2-2\rangle }}$
${\displaystyle \frac{GR(p^2,2)\left[u\right]}{\langle u^2-p\alpha \rangle }}$	${\displaystyle \frac{{\mathbb{F}}_2[u,v]}{\langle u^2,v^2\rangle }}$	${\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2,v^2\rangle }}$
${\displaystyle \frac{GR(p^2,2)\left[u\right]}{\langle u^2-p\rangle }}$	${\displaystyle \frac{GR(4,1)\left[u\right]}{\langle u^2-2-2u\rangle }}$	${\displaystyle \frac{GR(4,2)\left[u\right]}{\langle u^2-2-2u\rangle }}$
${\displaystyle \frac{GR(p^2,2)\left[u\right]}{\langle u^2\rangle }}$	${\displaystyle \frac{GR(4,1)\left[u\right]}{\langle u^2\rangle }}$	${\displaystyle \frac{GR(4,2)\left[u\right]}{\langle u^2\rangle }}$
$R^{*}={\displaystyle \frac{GR(p^2,2)\left[u\right]}{\langle u^3-p\beta ,pu\rangle }}$	${\displaystyle \frac{GR(4,1)\left[u\right]}{\langle u^2-2u\rangle }}$	${\displaystyle \frac{GR(4,2)\left[u\right]}{\langle u^2-2u\rangle }}$
${\displaystyle \frac{GR(p^2,2)[u,v]}{\langle u^2-p,v^2-p,uv,pu,pv\rangle }}$	${\displaystyle \frac{GR(4,1)[u,v]}{\langle u^2-p,v^2-p,uv,pu,pv\rangle }}$	${\displaystyle \frac{GR(4,2)[u,v]}{\langle u^2-p,v^2-p,uv,pu,pv\rangle }}$
${\displaystyle \frac{GR(p^2,2)[u,v]}{\langle u^2-p,v^2-p\alpha ,uv,pu,pv\rangle }}$	$R^{*}={\displaystyle \frac{GR(4,1)\left[u\right]}{\langle u^3-2\beta ,2u\rangle }}$	$R^{*}={\displaystyle \frac{GR(4,2)\left[u\right]}{\langle u^3-2\beta ,2u\rangle }}$
${\displaystyle \frac{GR(p^2,2)\left[u\right]}{\langle u^3-p,pu\rangle }}$	${\displaystyle \frac{GR(4,1)[u,v]}{\langle u^2,v^2,uv-2,2u,2v\rangle }}$	${\displaystyle \frac{GR(4,2)[u,v]}{\langle u^2,v^2,uv-2,2u,2v\rangle }}$
${\displaystyle \frac{GR(p^3,2)\left[u\right]}{\langle u^2-p^2\beta ,pu\rangle }}$	${\displaystyle \frac{{\mathbb{F}}_2\left[u\right]}{\langle u^4\rangle }}$	${\displaystyle \frac{{\mathbb{F}}_4\left[u\right]}{\langle u^4\rangle }}$
${\displaystyle \frac{GR(p^3,2)\left[u\right]}{\langle u^2-p^2,pu\rangle }}$	${\displaystyle \frac{GR(8,1)\left[u\right]}{\langle u^2-4,2u\rangle }}$	${\displaystyle \frac{GR(8,2)\left[u\right]}{\langle u^2-4,2u\rangle }}$
${\displaystyle \frac{{\mathbb{F}}_{p^2}\left[u\right]}{\langle u^4\rangle }}$

; those with 16 and 256 elements are particularly noteworthy. Note that, in the table, * means that there are exactly two copies of such rings.

The study is organized as follows: basic information regarding finite local rings and their modules is reviewed in Section 2. Consequently, we conclude our determination of all Frobenius local rings with $p^{4m}$ elements in Section 3. Of particular, those rings with $p^8$, 16, and 256 elements are provided. The section was divided into two subsections, $t=3$ and $t=4$, according to the values of the index of nilpotency of J. A few examples illustrate the main findings are provided in the section.

2. Preliminaries

This section addresses the basic information and notations that used in our analysis. In the discussion that follows, J is the radical of R, a finite local ring. Moreover, $R\left[x\right]$ means the ring of polynomials with coefficients in R, and $\langle y\rangle$ is denoted to the ideal generated by the element y of R. Regarding the outcomes stated here, we refer readers to [2,3,9,10,11,12,13].

Recall, in particular, that a ring R with a single maximum ideal J is referred to as local, and the ${\displaystyle \frac{R}{J}}={\mathbb{F}}_{p^m}\cong GF\left(p^m\right)$. The index of nilpotency of J is a positive integer t such that $J^t=0$ but $J^{t-1}\ne 0$. Now, when R is a finite local ring, by Nakayamas’ Lemma [12], then there exists ideals sequence,

$$R\supset J\supset J^2\supset \cdots \supset J^{t-1}\supset J^t=0.$$

(1)

Suppose that S is a ring and that M is an R-module. If $R\subseteq S$, then S is called an extension of R. Let I be an ideal of R, then the ideal $IS$ means the expansion of I to S. For any module M over R, the annihilator ideal is ${\mathrm{ann}}_R\left(M\right)=\{r:ra=0,a\in M\}$. Denote the set of all R-submodules of M by $I_R\left(M\right)$. The set of ideals of R, $I_R\left(M\right)$, if $M=R$, will be referred as $I\left(R\right)$. Moreover, the ${\displaystyle \frac{R}{I}}$-module on M has a natural structure for any ideal I of R that is contained in the annihilator of M. The scalar multiplication is $(r+I)a=ra$, where $r\in R$ and $a\in M$, so the lattice of R-submodules and ${\displaystyle \frac{R}{I}}$-submodules of A are the same. Suppose we have a strict inclusion sequence of R-submodules,

$$M=M_0\supset M_1\supset \cdots \supset M_{l-1}\supset M_l=0.$$

(2)

Such a chain is said to have length l, which is the number of joints. If all R-modules ${\displaystyle \frac{M_i}{M_{i+1}}}$ do not have a non-zero proper submodule or ${\displaystyle \frac{M_i}{M_{i+1}}}$ is a non-zero simple module, then the chain (2) is a composition series. A finite composition series for M may not exist, so the Jordan–Holder–Schreier Theorem [10] states that the length of a composition series for M, represented by $l_R\left(M\right)$, if it is finite or ∞, otherwise. Remember that these kinds of series have no relevance on the length $l_R\left(M\right)$. It follows that if $I=J$, then the dimension of a ${\displaystyle \frac{R}{J}}$-module is equal to that of the ${\mathbb{F}}_{p^m}$-vector space. Equations (1) and (2) implies that $t\le l_R\left(R\right)$, and if M admits a composition series over R, as in Equation (2), so ${\displaystyle \frac{M_i}{M_{i+1}}}\cong {\mathbb{F}}_{p^m}$ if they are considered as vector spaces over ${\mathbb{F}}_{p^m}$. Thus, as a direct result, we have

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

(3)

Suppose $R=M$, where R is a finite local ring, as direct consequences from Equation (3), there are l and m with order of J is $p^{(l-1)m}$ provided $J^l=0$, and the size of R, is $\left|R\right|=p^{lm}$. Also the characteristic of R is $p^n$, where $1\le n\le l$. In addition, R has a prime subring $R_0$ of the form $GR(p^n,m)$, called Galois ring with $p,n,m$. There are also $u_i\in J$, where $1\le i\le k$ with (as $R_0$-module)

$$\begin{array}{cc}\hfill & R=GR(p^n,m)+u_{1}GR(p^n,m)+u_{2}GR(p^n,m)+\cdots +u_{k}GR(p^n,m),\hfill \\ \hfill & J=pGR(p^n,m)+u_{1}GR(p^n,m)+u_{2}GR(p^n,m)+\cdots +u_{k}GR(p^n,m).\hfill \end{array}$$

(4)

The set of elements $\{u_1,u_2,\cdots ,u_k\}$ are generators of J [14].

A finite local ring R is Frobenius if its only minimal ideal is ${\mathrm{ann}}_R\left(J\right)$, which is simple. Equivalently, R is Frobenius if ${\displaystyle \frac{R}{J}}\cong \mathrm{soc}\left(R\right)$, where $\mathrm{soc}\left(R\right)$ is a direct sum of all minimal ideals of R. For Frobenius local rings, ${\mathrm{ann}}_R\left(J\right)=\mathrm{soc}\left(R\right)=J^{t-1}$ [3,4].

However, the converse is not true in general as we explained later. The family of Frobenius rings is large including finite chain and non-chain rings such as ${\displaystyle \frac{{\mathbb{F}}_p[u,v]}{\langle u^2,v^2\rangle }}$ and ${\displaystyle \frac{{\mathbb{Z}}_{p^n}\left[u\right]}{\langle u^2-p^{n-1},pu\rangle }}$, where $n\ge 3$. We recommend [2,7], for more examples.

3. Frobenius Local Rings of Order ${\mathit{p}}^{\mathbf{4}\mathit{m}}$

From now on, R is a Frobenius local ring with $p,n,m,4$ and k, then by taking different values of n and k, we shall prove, in this section, all possible structures of R and their enumeration. As $\mid R\mid =p^{4m}$, then R has a possible form as $R_0$—module of

$$R=\left\{\begin{array}{cc}GR(p,m)+u_{1}GR(p,m)+u_{2}GR(p,m)+u_{3}GR(p,m),\hfill & \mathrm{if}k=3,\hfill \\ GR(p^2,m)+uGR(p^2,m)+vGR(p^2,m),\hfill & \mathrm{if}k=2,\hfill \\ GR(p^3,m)+uGR(p^3,m),\hfill & \mathrm{if}k=1,\hfill \\ GR(p^4,m),\hfill & \mathrm{if}k=0,\hfill \end{array}\right.$$

(5)

where $GR(p,m)\cong {\mathbb{F}}_{p^m}$ and $n=3$. The group units of R, $U\left(R\right)$ is largely involved in the process of obtaining the construction of R. The structure of $U\left(GR(p^n,m)\right)$ has been completely captured in [15],

$$U\left(R\right)=\langle \alpha \rangle \times H,$$

where $H=1+J$. Suppose that $1\le i\le k$, and for every i, $t_i$ is the least positive integer satisfying $p^{t_i}{u}_i=0$. Since $\mid GR(p^n,m)\mid =p^{nm}$, thus

$$4=l=n+t_1+t_2+\cdots +t_k.$$

(6)

Moreover, since $1\le t_i\le n$ for each i. We shall observe that the shape and algebraic structure of R are entirely determined by the parameters $n,m,l,k$.

For the purpose of simplicity, we need to set $\alpha$ as a primitive element of ${\mathbb{F}}_{p^m}$ and

$$\Gamma \left(m\right)=\left(\alpha \right)\cup \left\{0\right\}=\{0,1,\alpha ,{\alpha }^2,\cdots ,{\alpha }^{p^m-2}\};$$

$$A=\{{\alpha }^{2i+1}:1\le i\le {\displaystyle \frac{p^m-1}{2}}-1\}=\{\alpha \in {\Gamma }^{*}\left(m\right):\alpha \notin {\left({\Gamma }^{*}\left(m\right)\right)}^2\};$$

$$B=\{{\alpha }^{2i}:1\le i\le {\displaystyle \frac{p^m-1}{2}}-1\}=\{\alpha \in {\Gamma }^{*}\left(m\right):\alpha \in {\left({\Gamma }^{*}\left(m\right)\right)}^2\}.$$

Next, we establish a very helpful relation among a local ring’s parameters $n,l,k$, which will reveal how to construct such a ring.

Proposition 1.

If R is a Frobenius local ring, then $J^{t-1}=$soc$\left(R\right)$.

Proof. 

As soc$\left(R\right)$ is principle, then so $J^{t-1}$ because $J^{t-1}\subseteq$ soc$\left(R\right)$. Furthermore, soc$\left(R\right)\cong {\displaystyle \frac{R}{J}}$ and $J^{t-1}\cong {\displaystyle \frac{R}{J}}$. Thus, we have $J^{t-1}=$soc$\left(R\right)$. □

The following example is very useful, since it shows that ann$\left(R\right)$ is not necessarily $J^{t-1}$ as it was claimed in [8]. In any way, $J^{t-1}\subseteq$ soc$\left(R\right)$. However, this is true when R is a Frobenius ring as in Proposition 1.

Example 1.

Let $R={\displaystyle \frac{{\mathbb{Z}}_{p^3}\left[u\right]}{\langle u^2,pu\rangle }}$. Then, $t_u=1$, and $t=3$. This implies, by Theorem 1, that R is not Frobenius. This is true because $J^2=\langle p^2\rangle$ while soc$\left(R\right)=\langle p^2,u\rangle$. This means, $J^2\subset$ soc$\left(R\right)$. This example shows also that the condition $u\notin$ann$\left(J\right)$ is necessary.

Suppose M be a R-module. A subset X of A spans M if its image $\overline{X}$ in ${\displaystyle \frac{M}{JM}}$ generates ${\displaystyle \frac{M}{JM}}$. A set of generators of A derived from lifting a basis of the ${\displaystyle \frac{R}{J}}$-vector space ${\displaystyle \frac{M}{JM}}$ is defined a minimal generating set for M over R and it is denoted by ${\nu }_R\left(M\right)$. Note that

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

(7)

If $M=J$, so

$$\nu \left(J\right)={\dim }_{{\mathbb{F}}_{p^m}}({\displaystyle \frac{J}{J^2}}).$$

Proposition 2.

Let $k=1$. Then,

$$t=\left\{\begin{array}{cc}n+i_p-1,\hfill & \mathrm{if}{t}_u\le n-1,\hfill \\ n+i_p,\hfill & \mathrm{if}{t}_u=n,\hfill \end{array}\right.$$

where $i_p$ is the largest number such that $p\in J^{i_p}$.

Proof. 

Assume that $k=1$, then $J=\langle p,u\rangle$. We consider two cases: When $\nu \left(J\right)=1;$$\nu \left(J\right)=2$. The case when $\nu \left(J\right)=1$, R is chain, and hence $t=n+i_p$. For the second case, non-chain, we have $p\in J\setminus J^2$, i.e., $i_p=1$. Thus, $J^i=\langle p^i,p^{i-1}u,u^i\rangle$. Since $u^i\in \langle p^i,p^{i-1}u\rangle$, $i\ge 2$ because $k=1$. Then, $J^i=\langle p^i,p^{i-1}u\rangle$. As $t\ge n$ and $J^n=\langle p^n,p^{n-1}u\rangle$, then $t=n$ if $t_u\le n-1$, and $t=n+1$, when $t_u=n$. □

The following results establish powerful tools in characterizing Frobenius local rings based just on their invariants $p,n,m,l,k$. First, we denote $t_u$ the least positive integer such that $p^{t_u}u=0$. Suppose that $k=1$, then $R=GR(p^n,m)+uGR(p^n,m)$. Also $u^2=p^d\beta h+p^{e}u{\beta }_1{h}_1$, where $\beta ,{\beta }_1\in {\Gamma }^{*}\left(m\right)$ and $h,h_1\in H$. Note that $u^2=0$ when $d=n$ and $e=t_u$.

Theorem 1.

assume R is a local non-chain ring with $k=1$. Then, R is Frobenius if and only if $u\notin$ann$\left(J\right)$ and $t_u\ne n-1$.

Proof. 

First, assume R is Frobenius, then $soc\left(R\right)$ is simple and unique. In addition, $soc\left(R\right)$ is principal, and hence we set $soc\left(R\right)=\langle w\rangle$. This implies that $pw=wu=0$. Because $w=r_0+r_{1}u$, then one can see that $r_0\in p^{n-1}GR(p^n,m)$ and $r_1\in p^{t_u-1}GR(p^n,m)$, and hence

$$w=p^{n-1}{v}_0+p^{t-1}{v}_{1}u,$$

(8)

where $v_0,v_1\in \Gamma \left(m\right)$ but not both 0. Subsequently, $w\in \langle p^{n-1},p^{t_u-1}u\rangle$. However, since $t-1⩽n-1$, then $soc\left(R\right)\subseteq \langle p^{t_u-1}\rangle$. Hence, $soc\left(R\right)=p^{n-1}R$. Now, we consider two cases to proceed. When $v_0=0$, so $soc\left(R\right)=\langle p^{t_u-1}u\rangle$, and hence $p^{n-1}u$ in the annihilator of J, and thus $p^{n-1}u\in soc\left(R\right)$, i.e., $n=t_u$. For the second direction, suppose that $v_0\ne 0$. Since $wu=0$ from Equation (8), $p^{n-1}u=0$ and $p^{t_u-1}{u}^2=0$, which leads to $p^{d+t_u-1}\beta h=0$. Thus, $t_u\ne n-1$ and $u\notin$ann$\left(J\right)$. Conversely, suppose $t_u\ne n-1$. Then, $p^{n-1}u\ne 0$ and $p^{n-1}uJ=0$. Consequently, $\mathrm{soc}\left(R\right)=\langle p^{n-1}u\rangle$ when $t_u=n$ and $\mathrm{soc}\left(R\right)=\langle p^{n-1}\rangle$ when $t_u<n-1$. Which means that R is Frobenius. □

Based on the previous results, we get

$$\mathrm{soc}\left(R\right)=\left\{\begin{array}{cc}\langle p^{n-1}\rangle ,\hfill & \mathrm{if}{t}_u<n-1,\hfill \\ \langle p^{n-1}u\rangle ,\hfill & \mathrm{if}{t}_u=n.\hfill \end{array}\right.$$

Proposition 3.

Suppose $l=4$ and $t=2$, then any local ring with such specifications is not Frobenius.

Proof. 

Note that, we have $k>1$, which means $\nu \left(J\right)>1$. Moreover, by Proposition 1, we get ann$\left(J\right)=J$ which is not principal since $J^2=0$. □

Proposition 4.

Let R be a local ring with $t=l$. Then, R is a chain ring. Moreover, $k=l-1$.

Proof. 

Let R be a local ring with invariants $p,n,m,k$. Since $J^l=0$ and $J^{l-1}\ne 0$. Then, consider the following chain

$$R=J^0\supset J\supset J^2\supset \cdots \supset J^{l-1}\supset J^l=0$$

is composition, and thus $\nu \left(J^i\right)=1$, where $1\le i\le l-1$. In particular, $\nu \left(J\right)$ must be 1 since $k\le l-1$. This indicates J is principal, and so R is chain, and $\nu \left(J^i\right)=1$ for all i. Therefore, $k=l-1$. □

3.1. Frobenius Local Non-Chain Rings of Order $p^{4m}$

In this subsection, we investigate Frobenius non-chain rings. In light of Proposition 4, we have $t<4$. As well by Proposition 3, the value of the index of nilpotency of our rings in this part is $t=3$.

Theorem 2.

Let R be a local ring with $l=4,t=k=3$ with $J=\langle u_1,u_2,u_3\rangle$. Then, R is Frobenius if and only if $u_i\notin \mathrm{ann}\left(J\right)$ for all i.

Proof. 

Since $k=3$ and $t=3$, then $n=1$. If R is Frobenius, then ann$\left(J\right)=J^2$. As $u_i{u}_j\in J^2$, then $J^2=\langle w\rangle$, where $w=u_3$, without loss of generality, is equal to $u_i{u}_j$, where $i,j=1,2$. This means $u_i^{\prime }$s are not in ann$\left(J\right)$. For the converse, since $J^2\subseteq$ ann$\left(J\right)$, then $u_i\notin J^2$ for some i, and thus $J^2$ must be generated by $w=u_i{u}_j$. Thus, ann$\left(J\right)=\langle w\rangle$, so R is Frobenius. □

We provide an example showing that the condition $u_i\notin$ann$\left(J\right)$ of Theorem 2 can not be relaxed.

Proposition 5.

Suppose $n=1$. Then, the generators of J satisfy only one of

$$\left\{\begin{array}{cc}\left\{\begin{array}{c}{u}^2=v^2\mathit{and}uv=0,\hfill \\ u^2=\alpha v^2\mathit{and}uv=0,\hfill \end{array}\right.\hfill & (p\ne 2)\hfill \\ \\ \left\{\begin{array}{c}{u}^2=v^2\mathit{and}uv=0,\hfill \\ u^2=v^2=0\mathit{and}uv\ne 0.\hfill \end{array}\right.\hfill & (p=2)\hfill \end{array}\right.$$

Proof. 

Since $n=1$, then $R_0={\mathbb{F}}_{p^m}$. Moreover, $k=3$, and hence $\nu \left(J\right)=2$ and $\nu \left(J^2\right)=1$. Thus, the generators of J are $u,v$ and w

$$R={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+w{\mathbb{F}}_{p^m},$$

$$J=u{\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+J^2,$$

$$J^2=w{\mathbb{F}}_{p^m},$$

where w is one of $u^2,v^2$ or $uv$. The case when $w=u^2$ is equivalent to that when $w=v^2$, and so there is only two cases. If $uv=0$, then $w=u^2$ and we get

$$u^2=\beta v^2,$$

where $\beta \in {\Gamma }^{*}\left(m\right)$. In fact, if $\beta =0$, then $u^2=0$, and thus soc$\left(R\right)=\langle u,v\rangle$ and this leads to R is not Frobenius. Now, if $p\ne 2$ and $\beta \in B$, then using the correspondence $u⟼u$ and $v⟼\sqrt{\beta }v$, we acquire $u^2=v^2$. While if $\beta \in A$, i.e., $\beta =\alpha$, then we get $u^2=\alpha v^2$. On the other hand, if $p=2$, then we have $\beta \in B$ in general. Thus, $u^2=v^2$. Finally, when $uv\ne 0$. If either $u^2=\beta uv$ or $v^2=\beta uv$, where $\beta \ne 0$, hence we set $u⟼u$ and $v⟼u-\beta v$ or $u⟼v-\beta u$ and $v⟼v$ and therefore we return to the first case. We suppose $u^2=v^2=0$, take $u⟼u+v$ and $v⟼u-v$ when $p\ne 2$, which reduces the situation to that when $uv=0$. When $p=2$, this is however not true, since $2\notin U\left(R\right)$ and all elements of $\langle u,v\rangle$ of ${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2,v^2\rangle }}$ have square zero and the elements u and v of ${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }}$ such that $u^2=v^2\ne 0$, therefore, ${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2,v^2\rangle }}\ncong {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }}$. □

Theorem 3.

Let R be a ring with $p^{4m}$ elements and characteristic p. Then, R is isomorphic to

$$\left\{\begin{array}{cc}\left\{\begin{array}{c}{\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }},\hfill \\ {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-\alpha v^2,uv\rangle }},\hfill \end{array}\right.\hfill & (\mathrm{if}p\ne 2)\hfill \\ \\ \left\{\begin{array}{c}{\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2-v^2,uv\rangle }},\hfill \\ {\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2,v^2\rangle }}.\hfill \end{array}\right.\hfill & (\mathrm{if}p=2.)\hfill \end{array}\right.$$

Proof. 

In the light of Proposition 5, we have if $p\ne 2$,

$$\begin{array}{c}\hfill u^2=v^2,uv=0,\\ \hfill u^2=\alpha v^2,uv=0.\end{array}$$

Thus, we have 2 rings with these characterizations, namely

$$\left\{\begin{array}{c}{\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }},\hfill \\ {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-\alpha v^2,uv\rangle }}.\hfill \end{array}\right.$$

Such rings are not isomorphic when $p\ne 2$ because if they are, we get $\alpha \in B$ which is impossible since $\langle \alpha \rangle ={\mathbb{F}}_{p^m}^{*}$ However, ${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }}\cong {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-\alpha v^2,uv\rangle }}$ is equivalent to $p=2$. Suppose that the two rings are isomorphic. There exist ${\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4\in \Gamma \left(m\right)$ such that $u_1={\delta }_{1}u+\delta u_{2}v$ and $v_1={\delta }_{3}u+{\delta }_{4}v$ are the minimal generators of J. These generators hold the relations of u and v, which result in equations of the form ${\beta }_1^2+{\beta }_2^2=\alpha ({\beta }_3^2+{\beta }_4^2)$, ${\beta }_3^2+{\beta }_4^2\ne 0$ and ${\beta }_1{\beta }_3+{\beta }_2{\beta }_4=0$ if and only if $p=2$. Let $p\ne 2$ and $uv\ne 0$, we replacing $u⟼u+v$ and $v⟼u-v$. After necessary calculations, we obtain $u^2=\beta v^2$, where $\beta =-1$. Thus, we return to the first case, and so ${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }}\cong {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2,v^2\rangle }}$ or ${\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-\alpha v^2,uv\rangle }}\cong {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2,v^2\rangle }}$ depending on $\beta$ is in B or in A, respectively. Assume $p=2$, then by Proposition 5, we have

$$\left\{\begin{array}{c}{\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2-v^2,uv\rangle }},\hfill \\ {\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2,v^2\rangle }}.\hfill \end{array}\right.$$

As we pointed out in the proof of the same proposition, these two rings are not isomorphic. In conclusion, we obtained 4 rings ($t=3$ and $n=1$),

$$\left\{\begin{array}{cc}\left\{\begin{array}{c}{\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }},\hfill \\ {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-\alpha v^2,uv\rangle }},\hfill \end{array}\right.\hfill & (\mathrm{if}p\ne 2)\hfill \\ \\ \left\{\begin{array}{c}{\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2-v^2,uv\rangle }},\hfill \\ {\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2,v^2\rangle }}.\hfill \end{array}\right.\hfill & (\mathrm{if}p=2)\hfill \end{array}\right.$$

□

Example 2.

Assume $R={\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{(u^3,v^2,uv)}}$. Since $v\in$ann$\left(J\right)$, then by Theorem 2, R is not Frobenius. Indeed, ann$\left(J\right)=\langle u^2,v\rangle$.

Proposition 6.

Suppose R is Frobenius local ring with $n=2$. Then,

$$\left\{\begin{array}{cc}{u}^2=p{\beta }_1,v^2={\beta }_{2}p\mathit{and}uv=pu=pv=0,\hfill & \mathrm{if}p\ne 2,\hfill \\ \left\{\begin{array}{c}{u}^2=v^2=2u=2v=0,uv\ne 0,\hfill \\ u^2=2,v^2=2,uv=2u=2v=0.\hfill \end{array}\right.\hfill & (\mathrm{if}p=2)\hfill \end{array}\right.$$

Proof. 

First, suppose $p\in J\setminus J^2$. Then, no such Frobenius rings because of completing squares will lead to $u^2=0$ when $p\ne 2$. Moreover, we have $u^2=2u$ if $p=2$. For such a case, soc$\left(R\right)$ is not simple, so R is not Frobenius. Secondly, if $p\in J^2$, which menas $\nu \left(J\right)=2$. As $J^2$ is cyclic, then $\nu \left(J^2\right)=1$ and so $J^2=\langle p\rangle$. As a result of that, we get

$$\begin{array}{cc}& R=GR(p^2,m)+uGR(p^2,m)+vGR(p^2,m),\hfill \\ & J=uGR(p^2,m)+vGR(p^2,m)+J^2,\hfill \\ & J^2=pGR(p^2,m).\hfill \end{array}$$

Since $p\in$ann$\left(J\right)$, then $pu=pv=0$. Now, if $p\ne 2$ and $uv\ne 0$, then if $u^2\ne 0$ or $v^2\ne 0$, we get $uv=0$ similar to Proposition 5. Moreover, if $u^2=v^2=0$, thus R is Frobenius if and only if $u^2\ne 0$ and $v^2\ne 0$. Suppose $p=2$, $u^2=v^2=2v=2u=0$ and $uv=2$. As $(2^m-1,2)=1$, $u^2=2$ and $v^2=2$. □

Theorem 4.

Let R be a Frobenius ring of order $p^{4m}$ and $n=2$. Then, R is

$$\left\{\begin{array}{cc}\left\{\begin{array}{c}{\displaystyle \frac{GR(p^2,m)[u,v]}{\langle u^2-p,v^2-p,uv,pu,pv\rangle }},\hfill \\ {\displaystyle \frac{GR(p^2,m)[u,v]}{\langle u^2-p\alpha ,v^2-p,pu,pv,uv\rangle }},\hfill \end{array}\right.\hfill & (\mathrm{if}p\ne 2)\hfill \\ \\ \left\{\begin{array}{c}{\displaystyle \frac{GR(4,m)[u,v]}{\langle u^2-2,v^2-2,2u,2v,uv\rangle }},\hfill \\ {\displaystyle \frac{GR(4,m)[u,v]}{\langle u^2,v^2,uv-2,2u,2v\rangle }}.\hfill \end{array}\right.\hfill & (\mathrm{if}p=2)\hfill \end{array}\right.$$

Proof. 

Observe that $u^2,v^2,uv\in J^2$. By Proposition 6, put

$$u^2=p{\beta }_1\mathit{and}{v}^2=p{\beta }_2,$$

in which ${\beta }_1,{\beta }_2\in \Gamma \left(m\right)$. In fact, if $uv=p\gamma$ and $\gamma \ne 0$, gives $uv=p$ by $u⟼{\gamma }^{-1}u$ and $v⟼v$. In addition, when $u^2=p{\beta }_1$ and $uv=p$, we have $uv=0$ by letting $v⟼u-{\beta }_{1}v$. Let $uv=p$ and ${\beta }_1={\beta }_2=0$. Substitute u with $u+v$ and $u-v$ with v, so $u^2=2p$, $v^2=-2p$, $uv=0$ provided that $p\ne 2$. So,

$${\displaystyle \frac{GR(p^2,m)[u,v]}{\langle u^2,v^2,uv-p,pu,pv\rangle }}\cong {\displaystyle \frac{GR(p^2,m)[u,v]}{\langle u^2-2p,v^2+2p,uv,pu,pv\rangle }}.$$

Hence, in general, $uv=0$. We conclude with the main case where ${\beta }_1$ and ${\beta }_2$ are in A. Given this, $u^2=\beta v^2$, where $\beta ={\beta }_1{\beta }_2^{-1}$, can be found in B. This is the same as having $u^2=p$ and $v^2=p{\beta }_2$, or having $u^2=p{\beta }_1$ and $v^2=p$. However, we have

$${\displaystyle \frac{GR(p^2,m)[u,v]}{\langle u^2-p{\beta }_1,v^2-p{\beta }_2,uv,pu,pv\rangle }}\cong {\displaystyle \frac{GR(p^2,m)[u,v]}{\langle u^2-p\alpha ,v^2-p,uv,pu,pv\rangle }}.$$

This means if ${\beta }_2\in B$, we get $R={\displaystyle \frac{GR(p^2,m)[u,v]}{\langle u^2-p\alpha ,v^2-p,uv,pu,pv\rangle }}$ and when ${\beta }_1\in B$, we have $R={\displaystyle \frac{GR(p^2,m)[u,v]}{\langle u^2-p\alpha ,v^2-p,uv,pu,pv\rangle }}$. If we let $u⟼v$ or $v⟼u$, then we get

$${\displaystyle \frac{GR(p^2,m)[u,v]}{\langle u^2-p,v^2-p\alpha ,uv,pu,pv\rangle }}\cong {\displaystyle \frac{GR(p^2,m)[u,v]}{\langle u^2-p\alpha ,v^2-p,uv,pu,pv\rangle }}.$$

Observe,

$$soc\left(R\right)=\left\{\begin{array}{cc}\langle u,p\rangle ,\hfill & \mathrm{if}\beta =0,\hfill \\ \langle v,p\rangle ,\hfill & \mathrm{if}\delta =0.\hfill \end{array}\right.$$

while when $p=2$, we have more copy,

$${\displaystyle \frac{GR(4,m)[u,v]}{\langle u^2,v^2,uv-2,2u,2v\rangle }}.$$

□

This example explains that if $v=u^2$ when $k=2$, R is not Frobenius.

Example 3.

Suppose that $R={\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^3,pu\rangle }}$. Hence, $t_u=1$, and $t=3$. This implies, by Theorem 1, that R is not Frobenius. In fact, $J^2=\langle u^2\rangle$ while soc$\left(R\right)=\langle p,u^2\rangle$. Which means that $J^2\subset$ soc$\left(R\right)$.

Proposition 7.

If $n=3$, then any Frobenius ring with $l=4$ and residue field ${\mathbb{F}}_{p^m}$ holds the following relation,

$$u^2=p^2\beta ,$$

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

Proof. 

The order R explains that $k=1$, $\nu \left(J\right)=2$ and also $\nu \left(J^2\right)=1$, which means

$$\begin{array}{cc}& R=GR(p^3,m)+uGR(p^3,m),\hfill \\ & J=uGR(p^2,m)+pGR(p^2,m)+J^2,\hfill \\ & J^2=p^{2}GR(p^2,m).\hfill \end{array}$$

Note that if $p\in J^2$, then $p^2\in J^4=0$, contradiction, since $n=3$. By $\mid R\mid =p^{4m}$, then $pu=0$. Also note that $u^2\in J^2$, and so $u^2=p^2\beta$, where $\beta \in \Gamma \left(m\right)$. Suppose $\beta =0$, then $u^2=0$, and hence $u\in$ann$\left(J\right)$ which is contradiction as R is Frobenius. Thus, $\beta \in {\Gamma }^{*}\left(m\right)$. When $p=2$, we get a similar relation, which is $u^2=4\beta$. □

Theorem 5.

If $n=3$, then any Frobenius ring with $l=4$ and residue field ${\mathbb{F}}_{p^m}$ is

$$\left\{\begin{array}{c}{\displaystyle \frac{GR(p^3,m)\left[u\right]}{\langle u^2-p^2,pu\rangle }},\hfill \\ {\displaystyle \frac{GR(p^3,m)\left[u\right]}{\langle u^2-p^2\alpha ,pu\rangle }}.\hfill \end{array}\right.$$

Proof. 

Since $n=3=t$, then we must have $k=1$ by Proposition 7. Now, If $p\in J^2$, then $p^2\in J^4=0$, which is impossible, and thus we have always $p\in J\setminus J^2$. Which means that $p,u$ are the minimal generating of J, i.e., $\nu \left(J\right)=2$, and so

$$\begin{array}{cc}& R=GR(p^3,m)+uGR(p^3,m),\hfill \\ & J=pGR(p^3,m)+uGR(p^3,m)+J^2,\hfill \\ & J^2=p^{2}GR(p^3,m).\hfill \end{array}$$

But since the order of R is $p^{4m}$, then $pu=0$. Furthermore, since R is Frobenius, then $u^2\ne 0$ because otherwise we would have $p^2$ and u are elements of the socle of R, and hence soc$\left(R\right)=\langle p^2,u\rangle$ is not simple. Thus, $u^2\in J^2$, and since soc$\left(R\right)=J^2$ is cyclic, therefore $J^2=\langle p^2\rangle =\langle u^2\rangle$. As a result, we have the following relations

$$\left\{\begin{array}{c}{u}^2=p^2\beta ,\hfill \\ pu=0,\hfill \end{array}\right.$$

where $\beta \in {\Gamma }^{*}\left(m\right)$. Hence, $R\cong {\displaystyle \frac{GR(p^3,m)\left[u\right]}{\langle u^2-p^2,pu\rangle }}$ or $R\cong {\displaystyle \frac{GR(p^3,m)\left[u\right]}{\langle u^2-p^2\alpha ,pu\rangle }}$. First note that there are ${\displaystyle \frac{GR(p^3,m)\left[u\right]}{\langle u^2-p^2,pu\rangle }}$ and ${\displaystyle \frac{GR(p^3,m)\left[u\right]}{\langle u^2-p^2\alpha ,pu\rangle }}$, which are not isomorphic, according to $\beta \in B$ and $\beta \in A$, respectively. Additionally, ${\displaystyle \frac{GR(p^3,m)\left[u\right]}{\langle u^2-p^2\alpha ,pu\rangle }}\cong {\displaystyle \frac{GR(p^3,m)\left[u\right]}{\langle u^2-p^2,pu\rangle }}$ if and only if $p=2$. In case when $p=2$, only one class of such ring exists, which is ${\displaystyle \frac{GR(8,m)\left[u\right]}{\langle u^2-4,2u\rangle }}$ since $(2^m-1,2)=1$, and hence $\sqrt{\beta }\in {\mathbb{F}}_{p^m}$. To conclude, we acquire 2 rings ($p\ne 2$) and one ring ($p=2$). □

3.2. Frobenius Local Chain Rings of Order $p^{4m}$

Remember that under set-theoretic inclusion, a finite ring R is termed to be a chain ring if its ideals is a unique sequence. The ring R is chain if and only if R is local and its maximum ideal J is primary, if and only if $t=l$. Being said that, we get $u_i=u^i$ for $i=1,2,\cdots ,k-1$. Moreover, $J=uR$, furthermore, it can be observed that $p\in J^k$, i.e., $u^k\in \langle p\rangle$. There is also $t^{\prime }$ with $1\le t^{\prime }\le k$ and $l=(n-1)k+t^{\prime }$. Note that,

$$u^k=p(r_0+r_{1}u+r_2{u}^2+...+r_{k-1}{u}^{k-1}),$$

where $r_0$ is a unit. This means u is a root of $g\left(x\right)$,

$$g\left(x\right)=x^k-p\sum _{i=0}^{k-1}{r}_i{x}^i.$$

(9)

The numbers p, n, m, l and k are linked with R,

$$R\cong {\displaystyle \frac{R_0\left[x\right]}{\langle g\left(x\right),p^{n-1}{x}^{t^{\prime }}\rangle }}.$$

(10)

According to [7], this extension is known as an Eisenstein extension of R over $R_0$. The element $f\in R\left[x\right]$ is referred as a basic irreducible if $\overline{f}$ is irreducible over ${\mathbb{F}}_{p^m}$. Now, assume $g\in R\left[x\right]$ is a basic irreducible polynomial and $m=\mathrm{deg}\left(\overline{g}\right)$, then, in this case, there exist a unit $\beta$ in $R\left[x\right]$ and a monic polynomial f in $R\left[x\right]$ satisfying $f=\beta g$ (see [10]). Now, if $J=\langle p\rangle$, we obtain $n=l$ and

$$R=R_0=GR(p^n,m)\cong {\mathbb{Z}}_{p^n}\left[\alpha \right]\cong {\displaystyle \frac{{\mathbb{Z}}_{p^n}\left[x\right]}{\langle g\left(x\right)\rangle }},$$

where $\alpha$ is a root of a basic irreducible polynomial $g\left(x\right)\in {\mathbb{Z}}_{p^n}\left[x\right]$ of degree m. The latter extension of ${\mathbb{Z}}_{p^n}$ is Galois.

Now, in the discussion will follow, we have $t=4=l$, then R is a chain ring, Proposition 4. In such case, $\nu \left(J\right)=1$, let $J=\langle u\rangle$. Consider the chain

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

We shall show by considering values for n, where $1\le n\le 4$. If $n=4$, then $p\in J\setminus J^2$, and hence $J=\langle u\rangle =\langle p\rangle$. Thus,

$$R=J^0\supset \langle p\rangle \supset \langle p^2\rangle \supset \langle p^3\rangle \supset \langle p^4\rangle =0.$$

Therefore, R is a Galois ring with

$$\begin{array}{cc}\hfill & R=GR(p^4,m),\hfill \\ \hfill & \mathrm{soc}\left(R\right)=\langle p^3\rangle .\hfill \end{array}$$

(11)

We have $GR(p^4,m)$ is Frobenius. Next, let $n=1$. As $k=3=l-1$ and $u^4=0$, then R is formed by

$$R={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+u^2{\mathbb{F}}_{p^m}+u^3{\mathbb{F}}_{p^m}.$$

There is one ring which is

$${\displaystyle \frac{{\mathbb{F}}_{p^m}\left[u\right]}{\langle u^4\rangle }}.$$

when $n=2$, we study two possible cases. The first theorem will handle the case $p\in J^2\setminus J^3$. While Theorem 7 investigates R when $p\in J^3$.

Theorem 6.

Suppose that R is Frobenius ring with $n=2$ and $p\in J^2\setminus J^3$. Then, the classes of R are given by

$$\begin{array}{ccc}{\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^2-p,p{u}^2\rangle }},& {\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^2-p\alpha ,p{u}^2\rangle }},& {\displaystyle \frac{GR(4,m)\left[u\right]}{\langle u^2-2-2u,2u^2\rangle }}.\end{array}$$

(12)

Proof. 

Put $n=2$, so $R_0=GR(p^2,m)$. Also, we get $k=1$, and therefore by [7],

$$\begin{array}{cc}& R=GR(p^2,m)+uGR(p^2,m),\hfill \\ & J=uGR(p^2,m)+J^2,\hfill \\ & J^2=pGR(p^2,m)+J^3,\hfill \\ & J^3=u^{3}GR(p^2,m).\hfill \end{array}$$

We have $u^2\in J=\langle p,u\rangle$, and it follows $u^2=p\beta +p{\beta }_{1}u$ since $pu\ne 0$, in the event of $\beta ,{\beta }_1\in \Gamma \left(m\right)$.

$$\begin{array}{c}\hfill R_{a_2,1}={\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^2-p\beta -p{\beta }_{1}u,p{u}^2\rangle }}.\end{array}$$

If ${\beta }_1\ne 0$ and $p\ne 2$, by enforcing square’s completing, we obtain $u^2=p\beta$. However, let $\beta =0$, then $u^2=0$ and $R\cong {\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^2\rangle }}$. The socle of R is $\langle u,pu\rangle$, thus it is not Frobenius. The case when $\beta \ne 0$, then

$$\left\{\begin{array}{cc}{\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^2-p\alpha ,p{u}^2\rangle }},\hfill & \mathrm{if}\beta \in A\hfill \\ {\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^2-p,p{u}^2\rangle }},\hfill & \mathrm{if}\beta \in B.\hfill \end{array}\right.$$

Thus, when $p\ne 2$, there are 2 classes. In contrast, when $p=2$ we get two identities which are $u^2=2u$ and $u^2=2+2u$. The first relation $u^2=2u$ is not chain. While when $\beta ={\beta }_1=0$, then $u^2=0$ and we are back to a studied case. As a summary,

$$\left\{\begin{array}{c}{\displaystyle \frac{GR(4,m)\left[u\right]}{\langle u^2-2,2u^2\rangle }},\hfill \\ {\displaystyle \frac{GR(4,m)\left[u\right]}{\langle u^2-2-2u,2u^2\rangle }}.\hfill \end{array}\right.$$

(13)

Henceforth,

$$soc\left(R\right)=\langle pu\rangle .$$

□

Theorem 7.

Given that R is a ring with $n=2$ and $p\in J^3$. So, R is

$$\begin{array}{cc}\hfill & R^{*}={\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^3-p\beta ,pu\rangle }},\hfill \\ \end{array}$$

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

Proof. 

Because $k=2$, then we have $t_1=t_2=1$, i.e., $pu=p{u}^2=0$. The direct sum form of R is

$$\begin{array}{cc}\hfill & R=GR(p^2,m)+uGR(p^2,m)+u^{2}GR(p^2,m),\hfill \\ \hfill & J=uGR(p^2,m)+J^2,\hfill \\ \hfill & J^2=u^{2}GR(p^2,m)+J^3,\hfill \\ \hfill & J^3=pGR(p^2,m).\hfill \end{array}$$

(14)

To characterize R, it should be first noted that R has $p,2,m,4$ as its parameters, see [2]. Observe $n=2$, and so $p\in J^3$, and thus $u^3=p\beta$, where $\beta \in \Gamma \left(m\right)$. If $\beta =0$, then we have $u^3=0$ and $pu=0$. Thus, R, with such specifications, is not Frobenius since soc$\left(R\right)=\langle p,u^2\rangle$ which is not cyclic. However, we go further with assumption that $\beta \in {\Gamma }^{*}\left(m\right)$, and hence we obtain $u^3=p\beta$ and $pu=0$. Therefore, R would be

$$R^{*}={\displaystyle \frac{{\mathbb{Z}}_{p^2}\left[u\right]}{\langle u^3-p\beta ,pu\rangle }},$$

(15)

where $\beta \in {\Gamma }^{*}\left(m\right)$. Based on results in [2], the number $N^{*}$ of $R^{*}$ has been already captured, which is

$$N^{*}=(p^m-1,3).$$

By other words, we acquire $(p^m-1,3)+1$ of these rings. □

Corollary 1.

There are $(p^m-1,3)+1$ of rings in Theorem 7. Also, soc$\left(R\right)=\langle p\rangle$, and so the rings are Frobenius.

Example 4.

The ring $R={\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^3,pu\rangle }}$ is not Frobenius because its socle is not simple. Note that $J=\langle p,u\rangle$, and also $pJ=u^{2}J=0$, which means that p and $u^2$ are in soc$\left(R\right)$, and thus soc$\left(R\right)=\langle p,u^2\rangle$.

However, when $n=3$, then $GR(p^3,m)\subseteq R$. As $J=\langle u\rangle$, then $p\notin J\setminus J^2$, otherwise $p^3\ne 0$ since $J^3\ne 0$. Now if $p\in J^2$, then $p^2\in J^4=0$, and thus $p^2=0$ which impossible since $n=3$. We deduce that n can not take 3, i.e., no chain rings with $t=4$ and $n=3$.

In summary, when $l=4$, we list all Frobeius chain and non-chain rings of order $p^{4m}$ in the following result.

Theorem 8.

If R is a Frobenius local ring of 4-length. Then, R is isomorphic to

• 1-Chain rings

  $$\begin{array}{ccc}{\displaystyle \frac{{\mathbb{F}}_{p^m}\left[u\right]}{\langle u^4\rangle }},\hfill & GR(p^4,m),\hfill & R^{*}={\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^3-p\beta ,pu\rangle }},\hfill \\ {\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^2-p\rangle }},\hfill & {\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^2-p\alpha \rangle }},\hfill & {\displaystyle \frac{GR(2^2,m)\left[u\right]}{\langle u^2-2-2u\rangle }}.\hfill \end{array}$$
• 2-Non-chain rings

  $$\begin{array}{ccc}{\displaystyle \frac{GR(4,m)\left[u\right]}{\langle u^2-2u\rangle }},\hfill & {\displaystyle \frac{GR(p^2,m)[u,v]}{\langle u^2-p,v^2-p,uv,pu,pv\rangle }},\hfill & {\displaystyle \frac{GR(p^2,m)[u,v]}{\langle u^2-p\alpha ,v^2-p,uv,pu,pv\rangle }},\hfill \\ {\displaystyle \frac{GR(4,m)[u,v]}{\langle u^2,v^2,uv-2,2u,2v\rangle }},\hfill & {\displaystyle \frac{GR(p^3,m)\left[u\right]}{\langle u^2-p\alpha ,pu\rangle }},\hfill & {\displaystyle \frac{GR(p^3,m)\left[u\right]}{\langle u^2-p,pu\rangle }},\hfill \\ {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-v^2,uv\rangle }},\hfill & {\displaystyle \frac{{\mathbb{F}}_{p^m}[u,v]}{\langle u^2-\alpha v^2,uv\rangle }},\hfill & {\displaystyle \frac{GR(p^2,m)\left[u\right]}{\langle u^2\rangle }},\hfill \\ {\displaystyle \frac{{\mathbb{F}}_{2^m}[u,v]}{\langle u^2,v^2\rangle }}.\hfill & \hfill \end{array}$$

As the primary outcomes of this section, we gather the previously established results in the theorem that follows.

Theorem 9.

The number of Frobenius local rings of length 4 is determined by

$$N_F=\left\{\begin{array}{c}\left\{\begin{array}{cc}12,\hfill & \mathrm{if}p\not\equiv 1\left(\mathrm{mod}3\right)\hfill \\ \\ 14,\hfill & \mathrm{if}p\equiv 1\left(\mathrm{mod}3\right).\hfill \end{array}\right.(p=2)\hfill \\ \\ \left\{\begin{array}{cc}13,\hfill & \mathrm{if}p\not\equiv 1\left(\mathrm{mod}3\right)\hfill \\ \\ 15,\hfill & \mathrm{if}p\equiv 1\left(\mathrm{mod}3\right).\hfill \end{array}\right.(p\ne 2)\hfill \end{array}\right.$$

The Table 1 gives a full classification of Frobenius local rings of length 4 and of order $p^8$, 16 and 256.

4. Conclusions

The importance of local rings with parameters $p,n,m,k$ in coding theory has made them an important research subject. Frobenius local rings have received particular attention because of their applicability in distance distributions and error-correcting process and symmetrized weight enumerators for linear codes and their duals. In this article, we looked into local rings of length 4, that is, rings where $l=4$. Up to isomorphism, we were able to successfully list every one of these rings with fixed invariants. Furthermore, based on their parameters, we categorized every Frobenius ring of these conditions in Theorems 8 and 9. In Table 1, we obtained the complete list of local rings of order $p^8$, 16, and 256 as a direct consequence. For potential future research, we propose exploring additional characterization of Frobenius rings and examining the algebraic structure of linear codes over them when l is significantly larger.