Commutative Chain Rings with Index of Nilpotency 5 and Residue Field ${\mathbb{F}}_{p^m}$

Abstract

This paper gives a thorough characterization of chain rings with index of nilpotency 5 and residue field ${\mathbb{F}}_{p^m},$ where p represents a prime number, contributing valuable insights to the field of algebraic structures. It carefully identifies and categorizes the family of chain rings with these specifications, thereby enhancing the understanding of their properties and applications. In addition, the work offers a detailed enumeration of all chain rings containing $p^{5m}$ elements. The significance of finite chain rings is emphasized, particularly in their suitability for coding theory, which confirms their relevance in contemporary mathematical and engineering contexts.

1. Introduction

Finite chain rings are an important category of rings within abstract algebra, possessing distinctive characteristics that attract substantial interest from mathematicians. By definition, a finite chain ring is a finite ring in which every ideal is included in a unique maximal principal ideal, resulting in a chain-like arrangement of ideals [1,2,3]. This stands in stark contrast to more general rings, which can have multiple maximal ideals leading to a more intricate structure of ideals. Basic examples of finite chain rings arise from Galois rings, first identified by Krull [4], particularly those of the form ${\mathbb{Z}}_{p^n}$, where p is a prime number. The exploration of finite chain rings is significantly linked to module theory, as finitely generated modules over these rings display well-defined behaviors that can help clarify the structure of the rings themselves [5,6]. Recently, it has been discovered that chain rings can serve as alphabets for linear codes, due to the applicability of two classical theorems by MacWilliams—the Extension Theorem and the MacWilliams Identities—to these rings, similar to their application in finite fields. For more on the topic, see [7,8,9,10,11,12,13].

Every finite commutative chain ring R can be represented as a quotient of a polynomial ring over its prime subring $R_0$, specifically

$$R\cong {\displaystyle \frac{R_0\left[u\right]}{I}},$$

(1)

where u is a generator of the maximal ideal J and I is an appropriate ideal formed by a combination of $u^{i_1}$ and $p^{i_2}$, with both $i_1$ and $i_2$ greater than 1. This connection allows for the use of techniques from algebraic geometry and number theory, enhancing the study of these rings. The classification of finite chain rings primarily involves analyzing the structure of the ideal I, leading to the conclusion that such rings can typically be expressed as a direct sum of the form

$$R=R_0+u{R}_0+\cdots +u^k{R}_0,$$

(2)

for some positive integer k. For some integer m, the residue field ${\displaystyle \frac{R}{J}}$ takes the form ${\mathbb{F}}_{p^m}$. The order of R is given by $|{\mathbb{F}}_{p^m}{|}^l={\left(p^m\right)}^l$, where l is the length of R when viewed as an R-module. Since R is finite, there exists a positive integer t such that $J^t=0$ and $J^{t-1}\ne 0$; this t is known as the index of nilpotency of J and satisfies $t\le l$. The additive order of 1 is $p^n$ for some n, referred to as the characteristic of R. In summary, the parameters p, n, m, k, and l associated with R are termed the invariants of R [14]. This paper aims to identify and classify all chain rings that have fixed invariants p, n, m, k, and $l=5$, specifically those of order $p^{5m}$. When $l=m=1$, it is well-established that there is only one chain ring, up to isomorphism, with p elements: the Galois field ${\mathbb{F}}_p$. For chain rings of order $p^2$, the options are ${\mathbb{F}}_{p^2}$ and ${\mathbb{Z}}_{p^2}$. If p is odd, the local rings with $lm=3$ include ${\mathbb{F}}_{p^3}$, ${\mathbb{Z}}_{p^3}$, ${\displaystyle \frac{{\mathbb{Z}}_{p^2}\left[u\right]}{⟨u^2-p,pu⟩}}$, and ${\displaystyle \frac{{\mathbb{Z}}_{p^2}\left[u\right]}{⟨u^2-p\alpha ,pu⟩}},$ where $\alpha$ is a primitive element of ${\mathbb{F}}_{p^m}.$ For $p=2$, local rings with $2^3$ elements include ${\mathbb{F}}_8$, ${\mathbb{Z}}_8$ and ${\displaystyle \frac{{\mathbb{Z}}_{2^2}\left[u\right]}{⟨u^2-2,2u⟩}},$ see [1]. Recently, the chain rings of order $p^4$ have been comprehensively detailed in several studies, see, for example, [10], where linear codes over such rings have garnered attention; while those with $m=1$ and $l=5$ have been studied in an earlier study by Corbas and Williams [15]. The current paper has two primary objectives: firstly, to construct chain rings with $m\ge 1$ and of length 5 (with a nilpotency index of 5) and, secondly, to classify and enumerate these rings under isomorphism, ensuring that they share the same invariants. In the discussion, all chain rings with $m=1$ are thoroughly listed and enumerated in Example 5.

The organization of the rest of this study is as follows: Section 2 reviews fundamental aspects of finite chain rings. Following that, Section 3 presents our findings regarding all chain rings with $p^{5m}$ elements, subdivided by the values of $n.$ Notably, we provide detailed information on those rings containing $p^5$ elements, supplemented with illustrative examples of our main results.

2. Preliminaries

In this section, we outline some fundamental concepts and notations that will be utilized throughout the manuscript. Let J denote the maximal ideal of R, a finite local chain ring with identity. For the results presented in this section, we refer to the literature [1,14,16,17,18,19,20].

A finite ring R is classified as a chain ring if the ideal lattice of R constitutes a chain. This condition is equivalent to R being local and having a primary maximal ideal J, or to the condition that the nilpotency index t of J equals l, where $t=l$ represents the length of the sequence

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

In this context, we observe that $J=uR$, and the quotient ${\displaystyle \frac{R}{J}}$ provides the residue field ${\mathbb{F}}_{p^m}$, which is isomorphic to a finite field $GF\left(p^m\right)$. Furthermore, there exists an integer k such that $p\in J^k$, meaning that $u^k\in ⟨p⟩$, where p is a prime number. The characteristic of the ring R takes the form $p^n$, for some positive integer n. Additionally, we can identify an integer e satisfying $1\le e\le k+1$ such that $l=(n-1)(k+1)+e$. Notably, such rings can generally be expressed as a direct sum of the form

$$R=R_0+u{R}_0+\cdots +u^k{R}_0,$$

leading to the equation

$$u^{k+1}=p(r_0+r_{1}u+r_2{u}^2+\cdots +r_k{u}^k),$$

where $r_0$ is a unit in R and $r_i\in R$. This relationship shows that u is a root of the Eisenstein polynomial given by

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

(3)

The parameters p, n, m, l, and k are termed the invariants of R. Generally, we have

$$R\cong {\displaystyle \frac{R_0\left[x\right]}{⟨g\left(x\right),p^{n-1}{x}^e⟩}}.$$

(4)

According to [1], this is recognized as an Eisenstein extension of R over $R_0$. The unit group $U\left(R\right)$ of the ring R is described in [16] and takes the form

$$U\left(R\right)=⟨\alpha ⟩\times H,$$

(5)

where $\alpha$ represents a primitive element from ${\mathbb{F}}_{p^m}$, and $H=1+J$. Based on Equation (5), we can express $u^{k+1},$ as

$$u^{k+1}=p\beta h,$$

with $\beta \in ⟨\alpha ⟩$ and $h\in H$.

An element $f\in R\left[x\right]$ is defined as basic irreducible if $\overline{f}$ is irreducible in ${\mathbb{F}}_{p^m}$. Suppose $g\in R\left[x\right]$ is a basic irreducible polynomial, and let $m=\mathrm{deg}\left(\overline{g}\right)$. In this scenario, there exists a unit $\beta$ in $R\left[x\right]$ and a monic polynomial f in $R\left[x\right]$ such that $f=\beta g$ (see [18]). If $J=⟨p⟩$, we then have $n=l$ and

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

where $\gamma$ is a root of a basic irreducible polynomial $g\left(x\right)\in {\mathbb{Z}}_{p^n}\left[x\right]$ of degree m. This extension of ${\mathbb{Z}}_{p^n}$ is a Galois extension. This category of rings is uniquely characterized by the parameters p, n, and m, and the group of automorphisms $\mathrm{Aut}\left(GR(p^n,m)\right)$ is cyclic of order m, generated by the Frobenius map $\rho$ [1].

The notations introduced in this section will remain applicable throughout the entirety of the manuscript.

3. Chain Rings of Length 5

In this section, let R be a local ring characterized by parameters $p,n,m,l$, and k. We will demonstrate all possible structures of R and their enumeration by varying the values of n and k.

For clarity, we define $\alpha$ as a primitive element of ${\mathbb{F}}_{p^m}$, leading to the sets:

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

(6)

To establish a relationship among the parameters t, n, and k, which is essential for proving the main results of this section, the following results are pivotal.

Proposition 1.

Let R be a finite local ring. Then, $t=l$ if and only if R is a chain ring with parameters $p,n,m,l$, and $k=t-1$.

Proof. 

Since $J^l=0$ and $J^{l-1}\ne 0$, the chain $R=J^0\supset J\supset J^2\supset \dots \supset J^{l-1}\supset J^l=0$ forms a composition series. Thus, we have ${\dim }_{{\mathbb{F}}_{p^m}}\left({\displaystyle \frac{J^i}{J^{i+1}}}\right)=1$ for $1\le i\le l-1$. In particular, ${\dim }_{{\mathbb{F}}_{p^m}}\left({\displaystyle \frac{J}{J^2}}\right)$ must equal 1, given that $k\le l-1$. This implies that J is principal, indicating that R is a chain ring with $k=l-1$. The converse is straightforward. □

We will show that the form and algebraic structure of R are completely determined by the parameters $p,n,m,l$, and k. Subsequently, we will establish a crucial relationship among the parameters n, l, and k of a chain ring, which will guide us in the construction of such rings.

Lemma 1.

Let R be a chain ring with parameters $p,n,m,l,$ and k. Then,

$$⌊{\displaystyle \frac{l}{n}}⌋-1\le k\le l-n.$$

(7)

Proof. 

Assume $p^{t_i}$ is the annihilator of $u^i$. Given that $l=n+t_1+t_2+\dots +t_k$ and $1\le t_i\le n$ for each i, we have $k\le t_1+t_2+\dots +t_k$. This leads us to conclude that $k+n\le l$, hence $k\le l-n$. Conversely, since there are k generators of R over $GR(p^n,m)$, any element a from $GR(p^n,m)$ can be expressed as

$$a=a_1+p{a}_1+\dots +p^{n-1}{a}_n,$$

where $a_i\in \mathrm{\Gamma }\left(m\right)$. Thus, there are $n+kn$ such generators over ${\mathbb{F}}_{p^m}$. Therefore, l cannot exceed $n+kn=(k+1)n$, which implies that $l\le (k+1)n$, yielding the result ${\displaystyle \frac{l}{n}}-1\le k$. □

It is pertinent to note the relationship among the invariants of R, expressed as

$$l=t=5=(k+1)(n-1)+e,$$

(8)

where $1\le e\le k+1$. We will investigate all feasible values for n. In particular, for $n=4$, the equation $5=3(k+1)+e$ has no valid solutions for k and e, indicating that no chain ring exists in this case. Consequently, this paper will be structured into subsections that focus on the values $n=1,2,3$, and 5.

3.1. Chain Rings of Characteristic p

This subsection assumes $n=1,$ and thus, by Lemma 1, we have $k=4=l-1.$ Moreover, the prime subring of R is a field of the form $R_0={\mathbb{F}}_{p^m}.$ The order of R is $p^{5m},$ and this forces the construction of R to be

$$\begin{array}{cc}\hfill & R={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+u^2{\mathbb{F}}_{p^m}+u^3{\mathbb{F}}_{p^m}+u^4{\mathbb{F}}_{p^m},\hfill \\ \hfill & J=u{\mathbb{F}}_{p^m}+J^2,\hfill \\ \hfill & J^2=u^2{\mathbb{F}}_{p^m}+J^3,\hfill \\ \hfill & J^3=u^3{\mathbb{F}}_{p^m}+J^4,\hfill \\ \hfill & J^4=u^4{\mathbb{F}}_{p^m},\hfill \\ \hfill & J^5=0.\hfill \end{array}$$

(9)

For general value of $k,$ R can be expanded as

$$R={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+\cdots +u^{k-1}{\mathbb{F}}_{p^m}.$$

(10)

It was found that this class of chain rings might be utilized to create new sequences with the best Hamming correlation properties, which is why it has been widely used in coding theory. These sequences are particularly valuable in applications such as spread spectrum communication, where they help mitigate interference and ensure reliable multiple access for users sharing the same frequency bandwidth. By creating sequences with optimal correlation properties, systems can enhance performance in environments with noise and other disturbances; for more details, see [10].

Theorem 1.

Every chain ring of order $p^{5m}$ and characteristic p is uniquely isomorphic to

$${\displaystyle \frac{{\mathbb{F}}_{p^m}\left[u\right]}{⟨u^5⟩}}.$$

(11)

Proof. 

The order of R explains that $k=4,$ and as such, if $J=⟨u⟩,$$u^5=0.$ As $n=1,$ we have $R_0={\mathbb{F}}_{p^m}.$ Hence, R is an isomorphic image of ${\mathbb{F}}_{p^m}\left[u\right]$ with $I=⟨u^5⟩.$ As we have seen earlier, $R\cong {\displaystyle \frac{{\mathbb{F}}_{p^m}\left[u\right]}{I}}.$ Therefore,

$$R\cong {\displaystyle \frac{{\mathbb{F}}_{p^m}\left[u\right]}{⟨u^5⟩}}.$$

(12)

□

Example 1.

Observe that Theorem 1 states that R is uniquely determined by the invariants $p,m,$ and $k.$ Which means any two rings R and T have, for example, $5,6,$ and $4$ as their fixed invariants; then, they are isomorphic,

$$R\cong T\cong {\displaystyle \frac{{\mathbb{F}}_{5^6}\left[u\right]}{⟨u^5⟩}}.$$

$$R={\mathbb{F}}_{5^6}+u{\mathbb{F}}_{5^6}+u^2{\mathbb{F}}_{5^6}+u^3{\mathbb{F}}_{5^6}+u^4{\mathbb{F}}_{5^6}.$$

If we denote $N(p,n=1,m,k)$, the number of non-isomorphic classes of chain rings of 5-length with invariants $p,n=1,m,k$, then, by Theorem 1, we obtain

$$N(p,n=1,m,k)=1.$$

3.2. Chain Rings of Characteristic $p^2$

Throughout, let R be a chain ring that has the characteristic $p^2,$ implying $n=2.$ The primary subring of R takes the type of $R_0=GR(p^2,m).$ The order of R and the implications of Lemma 1 indicate that $k=2.$ Consequently, the J under discussion has two scenarios depending on the position of p in the series

$$J\supset J^2\supset J^3\supset J^4\supset J^5=0.$$

First, observe that if $p\in J,$ then R does not form a chain since J is not principal, If $p\in J^2,$ then we reach a conclusion that $p^2\in J^4,$ which leads to $J^4=0$, which is not possible because $t=5.$ Therefore, we encounter two cases separately: $p\in J^3\setminus J^4$ (denoted as $i_p=3$) and $p\in J^4$ (denoted as $i_p=4$). Hence, we will analyze these cases in two theorems. The first theorem addresses the case where $p\in J^3,$ while the second theorem examines the case where $p\in J^4.$ However, in either case, we have

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

(13)

The ring R can be constructed in the form ${\displaystyle \frac{GR(p^2,m)\left[u\right]}{I}}.$ Our goal is to determine the precise structure of I in order to better understand the overall structure of $R,$ which relies heavily on the interactions between u and p given that $n=2.$ Furthermore, we aim to calculate the number $N(p,n=2,m,k=2)$ of chain rings that possess invariants $p,n=2,m,k=2.$

Theorem 2.

Every chain ring of order $p^{5m}$ and characteristic $p^2$ with $i_p=3$ is uniquely isomorphic to

$${\displaystyle \frac{GR(p^2,m)\left[u\right]}{⟨u^3-p\beta h,p{u}^2⟩}}.$$

(14)

Proof. 

We note first that $p\notin J^4.$ The order of R clarifies that k must be $2.$ Building on that, we have $J^3=pGR(p^2,m)+J^4$ and $J=uGR(p^2,m)+J^2,$ then multiplying gives $J^3\times J=J^4,$ and hence, $J^4=puGR(p^2,m)$, so that $J^3=pGR(p^2,m)+puGR(p^2,m).$ Thus, we obtain

$$\begin{array}{cc}\hfill & J=uGR(p^2,m)+J^2,\hfill \\ \hfill & J^2=u^{2}GR(p^2,m)+J^3,\hfill \\ \hfill & J^3=p{\mathbb{F}}_{p^m}+J^4,\hfill \\ \hfill & J^4=pu{\mathbb{F}}_{p^m},\hfill \\ \hfill & J^5=0.\hfill \end{array}$$

(15)

As $u^3\in J^3$ and thus we suppose $u^3=px+pyu,$ where x and y are in $GR(p^2,m).$ It is evident that $x\in U\left(GR(p^2,m)\right),$ because else $u^4=py{u}^2=0,$ which leads to $J^4=0$ and contradicts $t=5.$ By Equation (5), $x=\beta h_1,$ where $\beta \in {\mathrm{\Gamma }}^{*}\left(m\right)$ and $h_1\in H\left(GR(p^2,m)\right).$ This gives $u^3=p\beta h+pyu,$ and thus

$$u^3=p\beta h,$$

(16)

where $h=h_1+p{\beta }^{-1}yu\in U\left(R\right).$ The other relation between p and u is that $p{u}^2=0$ since $p{u}^2\in J^5.$ Note that here, $t^{\prime }=2=5-3(2-1).$ Therefore, R attains the construction of

$${\displaystyle \frac{GR(p^2,m)\left[u\right]}{⟨u^3-p\beta h,p{u}^2⟩}}.$$

(17)

Without difficulties, one can check that the ring R in Equation (17) has the right properties. To characterize such rings, we consider two cases. Case a. If $p\ne 3,$ we may have $H{\left(R\right)}^3=H\left(R\right)$ because $H\left(R\right)$ is p-group and the gcd of p and 3 is $(p,3)=1.$ As a result, we may replace h in Equation (16) by $h_0^3$ as the existence of $h_0$ is guaranteed by the order of $H\left(R\right).$ Hence, the equation will be reduced to that of

$$u^3=p\beta .$$

(18)

Let R and T be two rings with fixed invariants $p,n=2,m,k+1=3$ and let $u^3=p\beta ,v^3=p\gamma$ be their associated relations, respectively. Moreover, assume $\phi$ is an isomorphism, $R\to T,$ and $\sigma$ is its restriction to $GR(p^2,m).$ Observe, $\phi \left(u\right)=\delta hv,$ where $\delta \in {\mathrm{\Gamma }}^{*}\left(m\right)$ and $h\in H\left(T\right).$ So, we have

$$\begin{array}{c}\hfill p\sigma \left(\beta \right)=\phi \left(p\beta \right)=\phi \left(u^3\right)={\left(\phi \left(u\right)\right)}^3={\left(\delta hv\right)}^3={\delta }^3{h}^3{v}^3=p{\delta }^3{h}^3\gamma .\end{array}$$

Hence, $p\sigma \left(\beta \right)=p{\delta }^3{h}^3\gamma .$ Since $p{u}^2=0$ and $n=2,$ we may assume that $p{h}^3=p.$ By comparing sides, we obtain the relation

$$\sigma \left(\beta \right)={\delta }^3\gamma .$$

(19)

This relation classifies our considered rings. The latter equation can be interpreted as $\sigma \left(\beta \right){\gamma }^{-1}={\delta }^3,$ which means that $\sigma \left(\beta \right){\gamma }^{-1}\in {\displaystyle \frac{{\mathrm{\Gamma }}^{*}\left(m\right)}{{\mathrm{\Gamma }}^{*}{\left(m\right)}^3}}.$ As $p\ne 3,$ we obtain ${\mathrm{\Gamma }}^{*}{\left(m\right)}^3={\mathrm{\Gamma }}^{*}\left(m\right)$ when the order of ${\mathrm{\Gamma }}^{*}\left(m\right),$$p^m-1$ is relatively prime with $3,$ that is $(p^m-1,3)=1.$ In such cases, the quotient group is equal to the trivial group, i.e., ${\displaystyle \frac{{\mathrm{\Gamma }}^{*}\left(m\right)}{{\mathrm{\Gamma }}^{*}{\left(m\right)}^3}}=1,$ and hence, we obtain $\sigma \left(\beta \right)=\gamma .$ This means that $\gamma$ exists in the orbit of $\beta$ under the action of Aut$\left(GR(p^2,m)\right)$ on ${\mathrm{\Gamma }}^{*}\left(m\right).$ However, since $(p^m-1,3)=1$ and Aut$\left(GR(p^2,m)\right)$ is a cyclic group, the number of orbits N is

$$N={\displaystyle \frac{1}{m}}\sum _{i=0}^{m-1}(p^i-1,1)=1.$$

Then, the orbit of $\beta$ is o$\left(\beta \right)={\mathrm{\Gamma }}^{*}\left(m\right).$ Thus, whenever, $(p^m-1,3)=1,$ there is one and only class of such rings. This class is represented by

$${\displaystyle \frac{GR(p^2,m)\left[u\right]}{⟨u^3-p,p{u}^2⟩}}.$$

If $(p^m-1,3)=3,$ thus the order of the quotient group ${\displaystyle \frac{{\mathrm{\Gamma }}^{*}\left(m\right)}{{\mathrm{\Gamma }}^{*}{\left(m\right)}^3}}$ is $3,$ which means it is a cyclic group of order $3.$ However, for every element of ${\displaystyle \frac{{\mathrm{\Gamma }}^{*}\left(m\right)}{{\mathrm{\Gamma }}^{*}{\left(m\right)}^3}},$ we obtain a new class of chain rings associated with this element. In conclusion, there are 3 classes of such rings when $(p^m-1,3)=3.$ In summary, for Case a, the total number of chain rings with such properties is

$$N(p,n=2,m,3)=\left\{\begin{array}{cc}3,\hfill & \mathrm{if}p\equiv 1\left(\mathrm{mod}3\right),\hfill \\ 1,\hfill & \mathrm{if}p\not\equiv 1\left(\mathrm{mod}3\right).\hfill \end{array}\right.$$

Case b. If $p=3.$ We observe that $(p^m-1,3)=1,$ and thus, there is an element $\gamma$ of ${\mathrm{\Gamma }}^{*}\left(m\right)$ such that $\beta ={\gamma }^3.$ With this in mind, the relation in Equation (16) after replacing u by ${\gamma }^{-1}u$ becomes

$$u^3=ph,$$

(20)

where $h=1+\beta u\in H\left(R\right)$ and $\beta \in \mathrm{\Gamma }\left(m\right).$ To classify these rings, let R and T be two chain rings with such properties, and assume they have $u^3=ph$ and $v^3=p{h}_1,$, respectively. Suppose $R\cong T$ and $\phi$ is the isomorphism. There are $\delta \in {\mathrm{\Gamma }}^{*}\left(m\right)$ and $h_0\in H\left(T\right)$ such that $\varphi \left(u\right)=\delta h_{0}v.$ Note that

$$p(1+\sigma \left(\beta \right)\delta h_{0}v)=p\varphi (1+\beta u)=p\phi \left(h\right)=\phi \left(u^3\right)={\left(\phi \left(u\right)\right)}^3={\left(\delta h_{0}v\right)}^3=p{\delta }^3{h}_0^3{h}_1,$$

where $\sigma$ is the reduction of $\phi$ on $GR(p^2,m).$ As above, we may take $p{h}_0^3=p.$ Let $h_1=1+{\beta }_{1}v,$ where ${\beta }_1\in \mathrm{\Gamma }\left(m\right),$ then by comparison, we obtain

$${\delta }^3=1\mathrm{and}\sigma \left(\beta \right)={\delta }^3{\beta }_1.$$

Thus, $\sigma \left(\beta \right)={\beta }_1.$ Let Aut$\left(GR(p^2,m)\right)$ act on ${\mathrm{\Gamma }}^{*}\left(m\right),$ and thus, we have

$$N={\displaystyle \frac{1}{m}}\sum _{i=0}^{m-1}[(p^i-1,p^m-1)+1]={\displaystyle \frac{1}{m}}\sum _{i=0}^{m-1}{p}^{(i,m)},$$

(21)

where $(p^i-1,p^m-1)+1$ is the number of elements of ${\mathrm{\Gamma }}^{*}\left(m\right)$ fixed by ${\rho }^i$ and $0\le i\le m-1;$ noting that $(p^i-1,p^m-1)+1=p^{(i,m)}.$ In conclusion, when $p=3,$ there are ${\displaystyle \frac{1}{m}}{\sum }_{i=0}^{m-1}{p}^{(i,m)}$ of such rings. □

Remark 1.

It is worthy noting that when $(p^m-1,3)=1,$ then $R={\displaystyle \frac{GR(p^2,m)\left[u\right]}{⟨u^3-p\beta ,p{u}^2⟩}}\cong {\displaystyle \frac{GR(p^2,m)\left[u\right]}{⟨u^3-p,p{u}^2⟩}}$ if we take φ to be a correspondence that sends $u⟼{\beta }_{1}u,$ where ${\beta }^{-1}={\beta }_1^3.$ So,

$$\phi :\sum (x+uy+u^{2}z)⟼\sum (x+{\beta }_{1}uy+{\left({\beta }_{1}u\right)}^{2}z)$$

is indeed the desired isomorphism.

Example 2.

Let us assume that $m=1,$ then, by Equation (21), there are 3 classes of chain rings with invariants $p=3,n=2,m=1,k=2.$ These ring are listed as

$$\begin{array}{ccc}{\displaystyle \frac{{\mathbb{Z}}_9\left[u\right]}{⟨u^3-3,3u^2⟩}},& {\displaystyle \frac{{\mathbb{Z}}_9\left[u\right]}{⟨u^3-3-3u,3u^2⟩}},& {\displaystyle \frac{{\mathbb{Z}}_9\left[u\right]}{⟨u^3-3+3u,3u^2⟩}}.\end{array}$$

Theorem 3.

Every chain ring of order $p^{5m}$ and $n=2,i_p=4$ is uniquely isomorphic to

$${\displaystyle \frac{GR(p^2,m)\left[u\right]}{⟨u^4-p\beta ,pu⟩}}.$$

(22)

Proof. 

By the order of $R,$ we must then obtain $pu=0.$ Furthermore, $u^4=p\beta ,$ where $\beta \in {\mathrm{\Gamma }}^{*}\left(m\right).$ Furthermore, the order of R shows that $k=3,$ which implies that

$${\displaystyle \frac{GR(p^2,m)\left[u\right]}{⟨u^4-p\beta ,pu⟩}}.$$

(23)

Moreover, we have

$$\begin{array}{cc}\hfill & J=uGR(p^2,m)+J^2,\hfill \\ \hfill & J^2=u^{2}GR(p^2,m)+J^3,\hfill \\ \hfill & J^3=u^3{\mathbb{F}}_{p^m}+J^4,\hfill \\ \hfill & J^4=p{\mathbb{F}}_{p^m},\hfill \\ \hfill & J^5=0.\hfill \end{array}$$

(24)

As for existence, one can check easily that the latter ring is, in fact, a chain ring of order $p^{5m}$ of the type under consideration. In order to enumerate these rings up to isomorphism, let R and T be chain rings with the same invariants $p,n=2,m,k=3$ and with $u^4=p\beta$ and $v^4=p\gamma ,$ respectively. If we set $\phi$ as the isomorphism between R and T with $\phi \left(u\right)=\delta hv,$ where $\delta \in {\mathrm{\Gamma }}^{*}\left(m\right)$ and $h\in H\left(T\right).$ Thus, similar to the above discussion, we obtain

$$\sigma \left(\beta \right)={\delta }^4\gamma .$$

(25)

We first fix $p\ne 2.$ The latter relation can be read as $\sigma \left(\beta \right){\gamma }^{-1}\in {\displaystyle \frac{{\mathrm{\Gamma }}^{*}\left(m\right)}{{\mathrm{\Gamma }}^{*}{\left(m\right)}^4}}.$ The order of this quotient group determines the number of classes of chain rings with these properties. In fact, if $(p^m-1,4)=2,$ then, by a similar approach to that of proof of Theorem 2, there are $2N_0,$ where

$$N_0={\displaystyle \frac{1}{m}}\sum _{i=0}^{m-1}(p^i-1,2).$$

Since for $i\ge 1,$ we obtain $(p^m-1,2)=2,$ and then

$$N_0={\displaystyle \frac{2m-1}{m}}.$$

(26)

While if $(p^m-1,4)=4,$ thus, there are $4N_0$ of such classes, where N is

$$N_0={\displaystyle \frac{1}{m}}\sum _{i=0}^{m-1}(p^i-1,4).$$

Finally, if $p=2,$ hence $(2^m-1,4)=1,$ and thus, ${\displaystyle \frac{1}{m}}{\sum }_{i=0}^{m-1}(p^i-1,1)=1,$ it follows that there is a unique non-isomorphic class of these rings. To summarize, we have N of these rings in total, where

$$N=\left\{\begin{array}{c}\left\{\begin{array}{cc}{\displaystyle \frac{2(2m-1)}{m}},\hfill & \mathrm{if}{p}^m\equiv 3\left(\mathrm{mod}4\right),\hfill \\ {\displaystyle \frac{4}{m}}\sum _{i=0}^{m-1}(p^i-1,4),\hfill & \mathrm{if}{p}^m\equiv 1\left(\mathrm{mod}4\right),\hfill \end{array}\right.(p\ne 2)\hfill \\ 1, \mathrm{if}p=2.\hfill \end{array}\right.$$

(27)

□

Example 3.

Suppose $m=1.$ Thus, if we want to enumerate all chain rings having $p,n=2,m=1,k=3,$ by Equation (27), we obtain

$$N=\left\{\begin{array}{c}\left\{\begin{array}{cc}2,\hfill & \mathit{if}p\equiv 3\left(\mathit{mod}4\right),\hfill \\ 4,\hfill & \mathit{if}p\equiv 1\left(\mathit{mod}4\right),\hfill \end{array}\right.(p\ne 2)\hfill \\ 1, \mathit{if}p=2.\hfill \end{array}\right.$$

In the following, we show, by a different approach, the exact number of non-isomorphic classes of rings having a structure as in Theorem 3 when $p>2.$

Definition 1.

Let p be a fixed prime and let c be any positive integer. If $(c,p)=1,$ we define $\tau \left(c\right)$ to be the order of p in the group of units $U\left(c\right)$ of the ring of integers modulo $c.$ If $c=1,$ we set $\tau \left(c\right)=1.$ It is known that the order of $U\left(c\right)$ is $\varphi \left(c\right),$ where ϕ denotes the Euler ϕ-function. Therefore, $\tau \left(c\right)$ divides $\varphi \left(c\right).$

Theorem 4.

Let $p\ne 2$ and $d=(4,p^m-1).$ Then, there are exactly

$$\sum _{c|d}{\displaystyle \frac{\varphi \left(c\right)}{\tau \left(c\right)}}.$$

(28)

as many isomorphism classes of chain rings with invariants $p,2,m,5,3$.

Proof. 

Let ${\mathrm{\Gamma }}^{*}\left(m\right)$ denote the group of nonzero elements of ${\mathbb{F}}_{p^m}.$ For ${\mathrm{\Gamma }}^{*}\left(m\right),$ we define $\beta \sim \gamma$ if there exists $\delta$ and $\sigma \in Aut\left(R_0\right)$ such that

$$\sigma \left(\beta \right)={\beta }^{p^i}={\delta }^4\gamma .$$

(29)

has a solution in ${\mathrm{\Gamma }}^{*}\left(m\right)$ for some $0<i<m.$ Thus, it is sufficient to show that the number of equivalence classes is given by Formula (28) above. As is well known, ${\mathrm{\Gamma }}^{*}\left(m\right)$ is cyclic of order $q=p^m-1.$ Let $\alpha$ be a generator for ${\mathrm{\Gamma }}^{*}\left(m\right).$ It is straightforward to observe that an element in ${\mathrm{\Gamma }}^{*}\left(m\right)$ has a 4th root if and only if it is in the subgroup G generated by ${\alpha }^d$, where $f=(4,p^m-1).$ This leads to the conclusion that $\beta \sim \gamma$ if and only if ${\beta }^{p^t}G=\gamma G$ for some $1<t<m.$

To simplify the notation, we can replace ${\mathrm{\Gamma }}^{*}\left(m\right)$ with the additive group ${\mathbb{Z}}_{p^m-1}$ of integers modulo $p^m-1$. Thus, based on the previous notations, the equivalence relation can be rewritten as $j\sim i$ if and only if $j\equiv p^{t}imodd$ for some $0<t<m$. Let $s\left[i\right]$ denote the equivalence class of $i,$ and let $⟨f⟩=f{\mathbb{Z}}_q.$ Then, we have

$$s\left[i\right]={\cup }_{l=0}^{m-1}\left\{i{p}^l+⟨f⟩\right\}.$$

Let $(i,d)=e,$$c={\displaystyle \frac{f}{e}},$ and $\tau \left(c\right)=t.$ Then,

$$s\left[i\right]={\cup }_{l=0}^{t-1}\left\{i{p}^l+⟨f⟩\right\}.$$

The union in the equation above can be seen to be a disjoint union. Since $⟨f⟩$ has ${\displaystyle \frac{q}{f}}$ elements, we conclude that $s\left[i\right]$ has $\tau \left(c\right){\displaystyle \frac{q}{f}}$ elements. For $j\in s\left[i\right]$, we have $(i,f)=e$ if and only if $(j,f)=e.$ Thus, if we let $\ell \left(e\right)$ denote those $j\in {\mathbb{Z}}_q$ with $(j,f)=e,$ then $\ell \left(e\right)$ forms a union of equivalence classes, each having $\tau \left(c\right){\displaystyle \frac{q}{f}}$ elements. Conversely, $\ell \left(e\right)$ contains $\tau \left(c\right){\displaystyle \frac{q}{f}}$ elements, so it is clear that $\ell \left(e\right)$ is a union of ${\displaystyle \frac{\varphi \left(c\right)}{\tau \left(c\right)}}$ equivalence classes. As e varies over the divisors of f, so does $c={\displaystyle \frac{f}{e}}$ and vice versa. Consequently, Equation (28) provides the total number of classes as required. □

Remark 2.

It is well known that

$$\sum _{c|d}\varphi \left(c\right)=d.$$

(30)

So,

$$\sum _{c|d}{\displaystyle \frac{\varphi \left(c\right)}{\tau \left(c\right)}}\le d.$$

(31)

The equality holds if and only if $\tau \left(c\right)=1$ for all c, which occurs precisely when $d\left|\right(p-1).$

Remark 3.

Every finite chain ring with $(k+1,p)=1$ has an associated formula of the form $u^{k+1}=p\beta .$ To show this, let R be any finite chain ring. In the notation of 1.5, we have $R=R_0\left[u\right]$ where $u^{k+1}=py,$ where $y=a_0+a_{1}u+\cdots +a_k{u}^k$ with $a_i\in R_0,$ and $a_0$ being a unit in $R_0.$ We can express $y=a_0(1+u),$ where $u\in ⟨u⟩=J.$ Then, $w=1+u$ lies in the subgroup $1+J=H$ of the group of units of $R.$ Since J is an additive p-group, H forms a multiplicative p-group. Thus, if $(k+1,p)=1,$ which means that $k+1$ is relatively prime to the order of $H,$ guaranteeing that $w^{-1}$ has a kth root in H; let us denote this root by $v^{k+1}=w^{-1}.$ Let $\theta =uv,$ then $\theta =p{a}_0.$ Hence, the polynomial $x^{k+1}-p{a}_0$ must have a root in $R,$ indicating that R has such a relation between u and p.

Remark 4.

We have from Theorems 3 and 4,

$${\displaystyle \frac{1}{m}}\sum _{i=0}^{m-1}(p^i-1,4)=\sum _{c|d}{\displaystyle \frac{\varphi \left(c\right)}{\tau \left(c\right)}}.$$

(32)

3.3. Chain Rings of Characteristic $p^3$

For any chain ring with $n=3,$ its prime ring is of the form $GR(p^3,m).$ As $n=3$ and $l=5,$ then we must have $k=1.$ Hence,

$$R=GR(p^3,m)+uGR(p^3,m).$$

In this case, we have no choice but $p\in J^2,$ and thus $p^2\in J^4.$ Multiplying explains $J^3=puGR(p^3,m).$ It follows that

$$\begin{array}{cc}\hfill & J=uGR(p^3,m)+J^2,\hfill \\ \hfill & J^2=pGR(p^3,m)+J^3,\hfill \\ \hfill & J^3=puGR(p^3,m)+J^4,\hfill \\ \hfill & J^4=p^{2}GR(p^3,m),\hfill \\ \hfill & J^5=0.\hfill \end{array}$$

(33)

As $u^2\in J^2,$ then $u^2=px+puy,$ for some x and y in $GR(p^3,m).$ It is clear that $x\in U\left(GR(p^3,m)\right),$ because otherwise $u^2$ would be in $J^3.$ If $p\ne 2,$ one can complete the squares and then put $y=0.$ Hence, $u^2=p\beta h,$ where $\beta \in {\mathrm{\Gamma }}^{*}\left(m\right)$ and $h\in H.$ Furthermore, we have $p^{2}u=0$ as $\mid R\mid =p^{5m}.$ Note that $p{u}^2\ne 0,$ else $J^4=0.$

Theorem 5.

Every chain ring of order $p^{5m}$ and characteristic $p^3$ is uniquely isomorphic to

$${\displaystyle \frac{GR(p^3,m)\left[u\right]}{⟨u^2-p\beta h,p^{2}u⟩}}.$$

(34)

Proof. 

First, we assume $p\ne 2.$ Using the above discussion, we conclude that

$$u^2=p\beta h\mathrm{and}{p}^{2}u=0.$$

(35)

Therefore,

$$R\cong {\displaystyle \frac{GR(p^3,m)\left[u\right]}{⟨u^2-p\beta h,p^{2}u⟩}}.$$

One can check, as customary, that this quotient is in fact a ring of the right type. Now, we classify these rings up to isomorphism with invariants $p,n=3,m,k=1.$ Since $p\ne 2,$ then $(p,2)=1,$ and therefore the relation in (35) will be

$$u^2=p\beta \mathrm{and}{p}^{2}u=0.$$

By the latter equation, our rings are classified by the image of $\beta$ under the reduction ${\displaystyle \frac{{\mathrm{\Gamma }}^{*}\left(m\right)}{{\mathrm{\Gamma }}^{*}{\left(m\right)}^2}}.$ This means if $\beta \in B,$ then replacing u by $\sqrt{\beta }u$ gives

$$R\cong {\displaystyle \frac{GR(p^3,m)\left[u\right]}{⟨u^2-p,p^{2}u⟩}}.$$

While if $\beta \in A,$ then we obtain

$$R\cong {\displaystyle \frac{GR(p^3,m)\left[u\right]}{⟨u^2-p\alpha ,p^{2}u⟩}}.$$

These rings are different, and we only have ${\displaystyle \frac{GR(p^3,m)\left[u\right]}{⟨u^2-p\alpha ,p^{2}u⟩}}\cong {\displaystyle \frac{GR(p^3,m)\left[u\right]}{⟨u^2-p,p^{2}u⟩}}$ if $p=2.$ Now, let $p=2,$ then the relation between the generators is $u^2=2{\beta }_1+2u{\beta }_2,$ where ${\beta }_1\in {\mathrm{\Gamma }}^{*}\left(m\right)$ and ${\beta }_2\in \mathrm{\Gamma }\left(m\right).$ Since ${\beta }_1\in B$ because $(2^m-1,2)=1,$ then we may write the relation by $u^2=p+pu\gamma ,$ where $\gamma \in \mathrm{\Gamma }\mathrm{\Gamma }\left(m\right).$ If $\gamma =0,$ then we have 1 ring that is

$$R\cong {\displaystyle \frac{GR(8,m)\left[u\right]}{⟨u^2-2,4u⟩}}.$$

We assume $\gamma =\beta \ne 0,$

$$u^2=2+2u\beta .$$

(36)

A similar argument of Theorem 2 might be applied here and it gives,

$$N={\displaystyle \frac{1}{m}}\sum _{i=0}^{m-1}{2}^{(i,m)}.$$

(37)

In summary, there are 2 rings when $p\ne 2$ and for $p=2$ there are $N+1$ of such rings. □

Example 4.

Suppose $m=1,$ then if $p=2,$ we obtain $N=2$ by Equation (37). Thus, there are 3 rings with invariants $p=2,n=3,m=1,k=1,$ namely

$$\begin{array}{ccc}{\displaystyle \frac{{\mathbb{Z}}_8\left[u\right]}{⟨u^2-2,4u⟩}},& {\displaystyle \frac{{\mathbb{Z}}_8\left[u\right]}{⟨u^3-2-2u,4u⟩}},& {\displaystyle \frac{{\mathbb{Z}}_9\left[u\right]}{⟨u^3-2+2u,4u⟩}}.\end{array}$$

3.4. Chain Rings of Characteristic $p^5$

We investigate the case when $n=5.$ So, $n=l=5,$ and hence, $R_0=GR(p^5,m)$ the prime subring. Since, R and $R_0$ have the same order, thus

$$R=R_0=GR(p^5,m).$$

These rings are uniquely determined up to isomorphism with their invariants $p,n,m.$ Moreover,

$$\begin{array}{cc}\hfill & J=pGR(p^3,m)+J^2,\hfill \\ \hfill & J^2=p^{2}GR(p^3,m)+J^3,\hfill \\ \hfill & J^3=p^{3}GR(p^3,m)+J^4,\hfill \\ \hfill & J^4=p^{4}GR(p^3,m),\hfill \\ \hfill & J^5=0.\hfill \end{array}$$

(38)

Overall, every chain ring of order $p^{5m}$ and characteristic $p^5$ is uniquely isomorphic to $GR(p^5,m).$ This means that there is only one class of these rings. The construction of this ring is

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

where $g\left(x\right)$ is a basic irreducible polynomial over ${\mathbb{Z}}_{p^n},$ the ring of integers modulo $p^n.$ The element $\gamma$ is a root of $g\left(x\right).$ This extension is a Galois extension.

In the next example, we list all chain rings when $m=1.$

Example 5.

Let $m=1,$ then we present all chain rings with invariants $p,n,m=1,k$ as

$$\begin{array}{cccc}{\displaystyle \frac{{\mathbb{F}}_p\left[u\right]}{⟨u^5⟩}},& {\displaystyle \frac{{\mathbb{Z}}_9\left[u\right]}{⟨u^3-3,3u^2⟩}},& {\displaystyle \frac{{\mathbb{Z}}_9\left[u\right]}{⟨u^3-3-3u,3u^2⟩}},& {\displaystyle \frac{{\mathbb{Z}}_9\left[u\right]}{⟨u^3-3+3u,3u^2⟩}},\\ R^{*}={\displaystyle \frac{{\mathbb{Z}}_{p^2}\left[u\right]}{⟨u^3-p\beta ,p{u}^2⟩}},& R^{*}={\displaystyle \frac{{\mathbb{Z}}_{p^2}\left[u\right]}{⟨u^4-p\beta ,pu⟩}},& {\displaystyle \frac{{\mathbb{Z}}_{p^3}\left[u\right]}{⟨u^2-p,p^{2}u⟩}},& {\displaystyle \frac{{\mathbb{Z}}_{p^3}\left[u\right]}{⟨u^2-p\alpha ,p^{2}u⟩}},\\ {\displaystyle \frac{{\mathbb{Z}}_8\left[u\right]}{⟨u^2-2,4u⟩}},& {\displaystyle \frac{{\mathbb{Z}}_8\left[u\right]}{⟨u^3-2-2u,4u⟩}},& {\displaystyle \frac{{\mathbb{Z}}_8\left[u\right]}{⟨u^3-2+2u,4u⟩}},& {\mathbb{Z}}_{p^5}.\end{array}$$

Note that * means that there is more than one type of such ring according to certain conditions. Furthermore, their number is

$$N=\left\{\begin{array}{cc}1,\hfill & (n=1)\hfill \\ \left\{\begin{array}{c}\left\{\begin{array}{c}\left\{\begin{array}{cc}3,\hfill & \mathit{if}p\equiv 1\left(\mathit{mod}3\right),\hfill \\ 1,\hfill & \mathit{if}p\not\equiv 1\left(\mathit{mod}3\right),\hfill \end{array}\right.(p\ne 3)\hfill \\ \\ 3,\mathit{if}p=3\hfill \end{array}\right.(i_p=3)\hfill \\ \left\{\begin{array}{c}\left\{\begin{array}{cc}2,\hfill & \mathit{if}p\equiv 3\left(\mathit{mod}4\right),\hfill \\ 4,\hfill & \mathit{if}p\equiv 1\left(\mathit{mod}4\right),\hfill \end{array}\right.(p\ne 2)\hfill \\ 1,\mathit{if}p=2.\hfill \end{array}\right.(i_p=4)\hfill \end{array}\right.\hfill & (n=2)\hfill \\ \left\{\begin{array}{cc}2,\hfill & \mathit{if}p\ne 2,\hfill \\ 3,\hfill & \mathit{if}p=2,\hfill \end{array}\right.\hfill & (n=3)\hfill \\ 1.\hfill & (n=5)\hfill \end{array}\right.$$

This completes the classification of all chain rings of length 5 and of order $p^{5m}.$ Hence, all chain rings of order $p^{5m}$ and length 5 are classified in Table 1.

Table 1. Numbers of chain rings of length 5 and of order $p^{5m}$.

Char (R) $={\mathit{p}}^{\mathit{n}}$	Number of Non-Isomorphic Classes
$n=1$	1
$n=2$ and $i_p=3$	$$\left\{\begin{array}{c}\left\{\begin{array}{cc}3,\hfill & \mathrm{if}p\equiv 1(mod3),\hfill \\ 1,\hfill & \mathrm{if}p\not\equiv 1(mod3),(p\ne 3),\hfill \end{array}\right.\hfill \\ {\displaystyle \frac{1}{m}}\sum _{i=0}^{m-1}{3}^{(i,m)},\mathrm{if}p=3\hfill \end{array}\right.$$
$n=2$ and $i_p=4$	$$\left\{\begin{array}{c}\left\{\begin{array}{cc}{\displaystyle \frac{2(2m-1)}{m}},\hfill & \mathrm{if}{p}^m\equiv 3(mod4),\hfill \\ {\displaystyle \frac{4}{m}}\sum _{i=0}^{m-1}(p^i-1,4),\hfill & \mathrm{if}{p}^m\equiv 1(mod4),(p\ne 2),\hfill \end{array}\right.\hfill \\ 1,\hfill & \mathrm{if}p=2\hfill \end{array}\right.$$
$n=3$	$$\left\{\begin{array}{cc}2,\hfill & \mathrm{if}p\ne 2,\hfill \\ {\displaystyle \frac{1}{m}}\sum _{i=0}^{m-1}{2}^{(i,m)},\hfill & \mathrm{if}p=2\hfill \end{array}\right.$$
$n=4$	0
$n=5$	1

4. Conclusions

In conclusion, the study of local rings characterized by the parameters $p,n,m,l$, and k is vital to advancing coding theory. This article focused specifically on local rings of length 5, wherein, we systematically identified and classified these structures up to isomorphism, emphasizing their invariants. Notably, we organized the chain rings into a comprehensive table (Table 1) based on their parameters, thereby enhancing our understanding of their properties. Furthermore, we compiled a detailed list of all chain rings for the case where $m=1,$ contributing valuable insights to this intriguing area of research. It is important to emphasize that as the number k increases, the classification and structural difficulty of the associated rings also increase significantly. When the index of nilpotency exceeds $5,$ we believe that our existing tools may not suffice for classifying such rings, which indicates a need for the development of new techniques.