Classification of Finite Rings of Order 𝑝⁶ by Generators and Relations

Abstract

For any finite abelian group (𝑅,+), we define a binary operation or “multiplication” on 𝑅 and give necessary and sufficient conditions on this multiplication for 𝑅 to extend to a ring. Then we show when two rings made on the same group are isomorphic. In particular, it is shown that there are 𝑛+1 rings of order 𝑝^𝑛 with characteristic 𝑝^𝑛, where 𝑝 is a prime number. Also, all finite rings of order 𝑝⁶ are described by generators and relations. Finally, we give an algorithm for the computation of all finite rings based on their additive group.

1. Introduction

The problem of determining and classifying up to isomorphism finite rings has received considerable attention, both old and new, see [1–4]. It is well known that if 𝑅 is a finite ring (with identity), then the additive group of 𝑅 splits as the direct of its 𝑝-primary components 𝑅(𝑝), where 𝑝 is a prime number, and these are pairwise orthogonal ideals. Thus 𝑅 is the direct sum of the rings 𝑅(𝑝). We provide a new proof of this fact in the paper.

Now let 𝑝 be a prime number. It is easy to see that there are two rings of order 𝑝, ℤ_𝑝 and the null ring of order 𝑝. In [5], Raghavendran (1969) proved that there exist eleven rings of order 𝑝², only four of these rings have identity. Also in [6], Gilmer and Mott (1973) showed that there exist 4𝑝+48 rings of order 𝑝³(𝑝≠2), only twelve of these rings have identity, and 59 rings of order 8, only eleven of these rings have identity. For 𝑝⁴, a comprehensive list of noncommutative rings was first only drawn up in [7]. Commutative rings of order 𝑝⁴ have been characterized by Wilson [4]. Finally, Corbas and Williams (2000) in [1, 8], determined all rings of order 𝑝⁵. The rings of upper order are not still characterized.

Throughout the paper, all rings are associative (not necessarily commutative or with identity). For a set 𝑋, |𝑋| denotes the cardinal of 𝑋. For two integers 𝑎,𝑏 and positive integer 𝑛, we denote 𝑎≡_𝑛𝑏 in case 𝑎 and 𝑏 are congruent modulo 𝑛, and 𝑎∣𝑏 in case 𝑎 divides 𝑏. Also, for two integers 𝑎 and 𝑏, gcd(𝑎,𝑏) denotes the greatest common divisor of 𝑎 and 𝑏. In [9], the authors introduced a multiplication (similar to Theorem 3) on a finitely generated 𝑆-module to extend to an 𝑆-algebra, where 𝑆 is a commutative ring. Also a number of necessary and sufficient conditions for any 𝑆-module to extend to an 𝑆-algebra, where 𝑆 is a commutative ring, is given by Behboodi et al. [9]. Here instead of modules over a commutative ring, we focus our attention to the ℤ-modules. In Section 2, by using the method in [9], we define a binary operation or “multiplication" on a finite abelian group and extend to a ring. In particular, we prove that there are 𝑛+1 rings (up to isomorphism) of order 𝑝^𝑛 whose abelian group is cyclic. Also, all rings of order 𝑝⁶ whose abelian group is ℤ_𝑝⊕ℤ_𝑝5 are determined. In Section 3, an algorithm for the computation of all finite rings based on their additive group is given.

2. A Representation of Finite Rings and Fundamental Theorems

We begin the paper with the following well known fact and give a new proof of this fact.

Theorem 1. Let 𝑅 be a ring of order 𝑝^𝑛1₁𝑝^𝑛2₂⋯𝑝^𝑛𝑘_𝑘, where the 𝑝_𝑖 are distinct primes and the 𝑛_𝑖 are positive integers. Then 𝑅 is expressible, in a unique manner, as the direct sum of rings 𝑅_𝑖, where |𝑅_𝑖|=𝑝^𝑛𝑖_𝑖,𝑖=1,2,…,𝑘.

Proof. We know that the additive group 𝑅 is expressible, in a unique manner, as the direct sum of groups 𝑅_𝑖, where |𝑅_𝑖|=𝑝^𝑛𝑖_𝑖,𝑖=1,2,…,𝑘. We claim that 𝑅_𝑖 is an ideal of 𝑅, for 𝑖=1,…,𝑘. Suppose that 𝑎_𝑖,𝑏_𝑖∈𝑅_𝑖 and 𝑎_𝑖𝑏_𝑖=𝑥₁+𝑥₂+⋯+𝑥_𝑘 where 𝑥_𝑗∈𝑅_𝑗, for 1≤𝑗≤𝑘. Then 𝑎_𝑖𝑏_𝑖−𝑥_𝑖 = 𝑥₁ + ⋯ + 𝑥_𝑖−1 + 𝑥_𝑖+1 + ⋯ + 𝑥_𝑘. Since 𝑝^𝑛1₁ ⋯ 𝑝^𝑛𝑖−1_𝑖−1 𝑝^𝑛𝑖+1_𝑖+1 ⋯ 𝑝^𝑛𝑘_𝑘(𝑥₁+⋯+𝑥_𝑖−1+𝑥_𝑖+1+⋯+𝑥_𝑘)=0, 𝑝^𝑛1₁ ⋯ 𝑝^𝑛𝑖−1_𝑖−1 𝑝^𝑛𝑖+1_𝑖+1 ⋯ 𝑝^𝑛𝑘_𝑘(𝑎_𝑖𝑏_𝑖−𝑥_𝑖)=0. If 𝑎_𝑖𝑏_𝑖−𝑥_𝑖≠0, then we must have 𝑝_𝑖∣𝑝^𝑛1₁⋯𝑝^𝑛𝑖−1_𝑖−1𝑝^𝑛𝑖+1_𝑖+1⋯𝑝^𝑛𝑘_𝑘, a contradiction. Thus 𝑎_𝑖𝑏_𝑖−𝑥_𝑖=0 and so 𝑎_𝑖𝑏_𝑖∈𝑅_𝑖. Now suppose that 𝑎_𝑖∈𝑅_𝑖 and 𝑎_𝑗∈𝑅_𝑗 where 𝑖≠𝑗 and 1≤𝑖,𝑗≤𝑘. We show that 𝑎_𝑖𝑎_𝑗=0. Let 𝑎_𝑖𝑎_𝑗=𝑥₁+𝑥₂+⋯+𝑥_𝑘, where 𝑥_𝑡∈𝑅_𝑡, for 1≤𝑡≤𝑘. Then 𝑝^𝑛𝑖_𝑖𝑎_𝑖𝑎_𝑗=(𝑝^𝑛𝑖_𝑖𝑎_𝑖)𝑎_𝑗=0 implies that 𝑝^𝑛𝑖_𝑖(𝑥₁+𝑥₂+⋯+𝑥_𝑘)=0 and hence 𝑝^𝑛𝑖_𝑖𝑥_𝑡=0 for 1≤𝑡≤𝑘. Now for any 𝑡≠𝑖, if 𝑥_𝑡≠0, then 𝑝_𝑡∣𝑝^𝑛𝑖_𝑖, a contradiction. Thus for all 𝑡≠𝑖, 𝑥_𝑡=0. On the other hand, since 𝑝^𝑛𝑗_𝑗𝑎_𝑖𝑎_𝑗=𝑎_𝑖(𝑝^𝑛𝑗_𝑗𝑎_𝑗)=0, by a similar argument, we conclude that for all 𝑡≠𝑗,𝑥_𝑡=0. Thus 𝑥₁=𝑥₂=⋯=𝑥_𝑘=0 and so for all 1≤𝑖≤𝑘, 𝑅_𝑖 is an ideal of 𝑅, and the proof is complete.

Remark 2. If 𝑅 is a finite ring, then its additive group is a finite abelian group and is thus a direct product of cyclic groups. Suppose these have generators 𝑎₁,𝑎₂,…,𝑎_𝑛 of orders 𝑚₁,𝑚₂,…,𝑚_𝑛. Then the ring structure is determined by the 𝑛² products 𝑎_𝑖𝑎_𝑗=^𝑛_𝑘=1𝑤_𝑖𝑗𝑘𝑎_𝑘with𝑤_𝑖𝑗𝑘∈ℤ (1) and thus by the 𝑛³ structure constants 𝑤_𝑖𝑗𝑘. As [6] we introduce a convenient notation, motivated by group theory, for giving the structure of a finite ring. A presentation for a finite ring 𝑅 consists of a set of generators 𝑎₁,𝑎₂,…,𝑎_𝑛 of the additive group of 𝑅 together with relations. The relations are of two types: (i)𝑚_𝑖𝑎_𝑖=0 for 𝑖=1,…,𝑛; (ii)𝑎_𝑖𝑎_𝑗=∑^𝑛_𝑘=1𝑤_𝑖𝑗𝑘𝑎_𝑘 with 𝑤_𝑖𝑗𝑘∈ℤ for 𝑖,𝑗=1,…,𝑛.
If the ring 𝑅 has the presentation above we write 𝑅=𝑎₁,…,𝑎_𝑛∶𝑚_𝑖𝑎_𝑖=0,𝑎_𝑖𝑎_𝑗=^𝑛_𝑘=1𝑤_𝑖𝑗𝑘𝑎_𝑘,𝑖,𝑗=1,…,𝑛. (2)

Theorem 3. Let 𝑅 be a finite ring with additive group ⊕^𝑛_𝑖=1𝐴_𝑖, where 𝐴_𝑖=⟨𝑎_𝑖⟩ (1≤𝑖≤𝑛) are cyclic subgroups of orders 𝑚_𝑖, respectively. Assume 𝑤_𝑖𝑗𝑘∈ℤ, 𝜆_𝑖𝑗=(𝑚_𝑖,𝑚_𝑗), where 1≤𝑖,𝑗,𝑘≤𝑛, and “⋅” is the following operation ⋅∶𝑅×𝑅→𝑅_𝑛𝑖=1𝑠_𝑖𝑎_𝑖,^𝑛_𝑗=1𝑡_𝑗𝑎_𝑗→^𝑛_𝑘=1^𝑛_𝑖,𝑗=1𝑠_𝑖𝑡_𝑗𝑤_𝑖𝑗𝑘𝑎_𝑘. (3) Then (1)“⋅” is well-defined if and only if 𝑚_𝑘∣𝜆_𝑖𝑗𝑤_𝑖𝑗𝑘∀1≤𝑖,𝑗,𝑘≤𝑛; (∗)(2)“⋅” is always distributive, in the case when “⋅” is well-defined;(3)“⋅” is associative if and only if ^𝑛_𝛼=1𝑤_𝑖𝑗𝛼𝑤_𝛼𝑙𝑘≡_𝑚𝑘^𝑛_𝛼=1𝑤_𝑗𝑙𝛼𝑤_𝑖𝛼𝑘∀1≤𝑖,𝑗,𝑘,𝑙≤𝑛. (∗∗)Consequently, 𝑎₁,…,𝑎_𝑛∶𝑚_𝑖𝑎_𝑖=0,𝑎_𝑖𝑎_𝑗=^𝑛_𝑘=1𝑤_𝑖𝑗𝑘𝑎_𝑘,𝑖,𝑗=1,…,𝑛 (4) is a ring if and only if (∗) and (∗∗) above hold.

Proof. (1) (⇒). Assume “⋅” is well-defined and fix 𝑖,𝑗 where 1≤𝑖,𝑗≤𝑛. Let 𝑥∈𝑚_𝑖ℤ. Then by definition of “⋅”, we have ^𝑛_𝑘=1𝑥𝑤_𝑖𝑗𝑘𝑎_𝑘=𝑥𝑎_𝑖⋅𝑎_𝑗=(0)⋅𝑎_𝑗=0. (5) Thus for each 𝑘=1,…,𝑛, we deduce 𝑥𝑤_𝑖𝑗𝑘𝑎_𝑘=0thatis,𝑚_𝑖𝑤_𝑖𝑗𝑘ℤ⊆𝑚_𝑘ℤ, for 𝑘=1,…,𝑛. Similarly, 𝑚_𝑗𝑤_𝑖𝑗𝑘ℤ⊆𝑚_𝑘ℤ, for 𝑘=1,…,𝑛. Therefore, 𝑚_𝑖ℤ+𝑚_𝑗ℤ𝑤_𝑖𝑗𝑘⊆𝑚_𝑘ℤ∀1≤𝑖,𝑗,𝑘≤𝑛. (6) Now one can easily check that the last relation holds if and only if (∗) holds.
(1) (⇐). Assume ^𝑛_𝑖=1𝑠_𝑖𝑎_𝑖=^𝑛_𝑖=1𝑢_𝑖𝑎_𝑖,^𝑛_𝑗=1𝑡_𝑗𝑎_𝑗=^𝑛_𝑗=1𝑣_𝑗𝑎_𝑗, (7) where 𝑠_𝑖,𝑢_𝑖,𝑡_𝑗,𝑣_𝑗∈ℤ. Then 𝑠_𝑖𝑎_𝑖=𝑢_𝑖𝑎_𝑖 and 𝑡_𝑗𝑎_𝑗=𝑣_𝑗𝑎_𝑗 for all 𝑖,𝑗. Let 𝑥_𝑖=𝑢_𝑖−𝑠_𝑖 and 𝑦_𝑗=𝑣_𝑗−𝑡_𝑗. Then 𝑢_𝑖=𝑥_𝑖+𝑠_𝑖, 𝑣_𝑗=𝑦_𝑗+𝑡_𝑗, 𝑥_𝑖∈𝑚_𝑖ℤ and 𝑦_𝑗∈𝑚_𝑗ℤ. Hence 𝑢_𝑖𝑣_𝑗−𝑠_𝑖𝑡_𝑗=𝑠_𝑖𝑦_𝑗+𝑥_𝑖𝑡_𝑗+𝑥_𝑖𝑦_𝑗=𝑥_𝑖𝑡_𝑗+𝑠_𝑖+𝑥_𝑖𝑦_𝑗∈𝑚_𝑖ℤ+𝑚_𝑗ℤ (8) and so 𝑢_𝑖𝑣_𝑗−𝑠_𝑖𝑡_𝑗𝑤_𝑖𝑗𝑘∈𝑚_𝑖ℤ+𝑚_𝑗ℤ𝑤_𝑖𝑗𝑘∀𝑘(1≤𝑘≤𝑛). (9) Now by condition (∗) we have 𝑢_𝑖𝑣_𝑗−𝑠_𝑖𝑡_𝑗𝑤_𝑖𝑗𝑘𝑎_𝑘=0∀𝑘(1≤𝑘≤𝑛). (10) It follows that ^𝑛_𝑘=1^𝑛_𝑖,𝑗=1𝑠_𝑖𝑡_𝑗𝑤_𝑖𝑗𝑘𝑎_𝑘=^𝑛_𝑘=1^𝑛_𝑖,𝑗=1𝑢_𝑖𝑣_𝑗𝑤_𝑖𝑗𝑘𝑎_𝑘, (11) that is, ^𝑛_𝑖=1𝑠_𝑖𝑎_𝑖⋅^𝑛_𝑗=1𝑡_𝑗𝑎_𝑗=^𝑛_𝑖=1𝑢_𝑖𝑎_𝑖⋅^𝑛_𝑗=1𝑣_𝑗𝑎_𝑗. (12) Thus the operation “⋅” is well-defined.
(2) Suppose the operation “⋅” is well-defined. Then ^𝑛_𝑖=1𝑠_𝑖𝑎_𝑖+^𝑛_𝑖=1𝑢_𝑖𝑎_𝑖⋅^𝑛_𝑗=1𝑡_𝑗𝑎_𝑗=^𝑛_𝑖=1𝑠_𝑖+𝑢_𝑖𝑎_𝑖⋅^𝑛_𝑗=1𝑡_𝑗𝑎_𝑗=^𝑛_𝑘=1^𝑛_𝑖,𝑗=1𝑠_𝑖+𝑢_𝑖𝑡_𝑗𝑤_𝑖𝑗𝑘𝑎_𝑘=^𝑛_𝑘=1^𝑛_𝑖,𝑗=1𝑠_𝑖𝑡_𝑗𝑤_𝑖𝑗𝑘𝑎_𝑘+^𝑛_𝑘=1^𝑛_𝑖,𝑗=1𝑢_𝑖𝑡_𝑗𝑤_𝑖𝑗𝑘𝑎_𝑘=^𝑛_𝑖=1𝑠_𝑖𝑎_𝑖⋅^𝑛_𝑗=1𝑡_𝑗𝑎_𝑗+^𝑛_𝑖=1𝑢_𝑖𝑎_𝑖⋅^𝑛_𝑗=1𝑡_𝑗𝑎_𝑗. (13)
On the other hand ^𝑛_𝑖=1𝑠_𝑖𝑎_𝑖⋅^𝑛_𝑗=1𝑡_𝑗𝑎_𝑗+^𝑛_𝑖=1𝑣_𝑗𝑎_𝑗=^𝑛_𝑖=1𝑠_𝑖𝑎_𝑖⋅^𝑛_𝑗=1𝑡_𝑗+𝑣_𝑗𝑎_𝑗=^𝑛_𝑘=1^𝑛_𝑖,𝑗=1𝑠_𝑖𝑡_𝑗+𝑣_𝑗𝑤_𝑖𝑗𝑘𝑎_𝑘=^𝑛_𝑘=1^𝑛_𝑖,𝑗=1𝑠_𝑖𝑡_𝑗𝑤_𝑖𝑗𝑘𝑎_𝑘+^𝑛_𝑘=1^𝑛_𝑖,𝑗=1𝑠_𝑖𝑣_𝑗𝑤_𝑖𝑗𝑘𝑎_𝑘=^𝑛_𝑖=1𝑠_𝑖𝑎_𝑖⋅^𝑛_𝑗=1𝑡_𝑗𝑎_𝑗+^𝑛_𝑖=1𝑠_𝑖𝑎_𝑖⋅^𝑛_𝑗=1𝑣_𝑗𝑎_𝑗. (14) Thus “⋅” is a distributive operation.
(3) By definition of “⋅”, we have 𝑎_𝑖⋅𝑎_𝑗=^𝑛_𝑘=1𝑤_𝑖𝑗𝑘𝑎_𝑘∀1≤𝑖,𝑗≤𝑛 (15) and so 𝑎_𝑖⋅𝑎_𝑗⋅𝑎_𝑙=^𝑛_𝛼=1𝑤_𝑖𝑗𝛼𝑎_𝛼⋅𝑎_𝑙=^𝑛_𝑘=1^𝑛_𝛼=1𝑤_𝑖𝑗𝛼𝑤_𝛼𝑙𝑘𝑎_𝑘 (16) and also 𝑎_𝑖⋅𝑎_𝑗⋅𝑎_𝑙=𝑎_𝑖⋅^𝑛_𝛼=1𝑤_𝑗𝑙𝛼𝑎_𝛼=^𝑛_𝑘=1^𝑛_𝛼=1𝑤_𝑗𝑙𝛼𝑤_𝑖𝛼𝑘𝑎_𝑘. (17) Now, it is clear that “⋅” is associative if and only if (𝑎_𝑖⋅𝑎_𝑗)⋅𝑎_𝑙=𝑎_𝑖⋅(𝑎_𝑗⋅𝑎_𝑙) for all 1≤𝑖,𝑗,𝑙≤𝑛, if and only if ^𝑛_𝛼=1𝑤_𝑖𝑗𝛼𝑤_𝛼𝑙𝑘≡_𝑚𝑘^𝑛_𝛼=1𝑤_𝑗𝑙𝛼𝑤_𝑖𝛼𝑘∀1≤𝑖,𝑗,𝑘,𝑙≤𝑛⋅ (18)

Theorem 4. For 𝑛∈ℕ, let 𝑅₁=𝑎₁,…,𝑎_𝑛∶𝑚_𝑖𝑎_𝑖=0,𝑎_𝑖𝑎_𝑗=^𝑛_𝑘=1𝑤_𝑖𝑗𝑘𝑎_𝑘,𝑅,𝑖,𝑗=1,…,𝑛₂=𝑎₁,…,𝑎_𝑛∶𝑚_𝑖𝑎_𝑖=0,𝑎_𝑖𝑎_𝑗=^𝑛_𝑘=1𝑧_𝑖𝑗𝑘𝑎_𝑘,𝑖,𝑗=1,…𝑛 (19) be two presentations with suitable 𝑤_𝑖𝑗𝑘,𝑧_𝑖𝑗𝑘∈ℤ, (1≤𝑖,𝑗,𝑘≤𝑛). Assume 𝑝_𝑖𝑗∈ℤ where 1≤𝑖,𝑗≤𝑛 and 𝑓 is the following map: 𝑓∶𝑅₁→𝑅₂,𝑓^𝑛_𝑖=1𝑠_𝑖𝑎_𝑖=^𝑛_𝑗=1^𝑛_𝑖=1𝑠_𝑖𝑝_𝑖𝑗𝑏_𝑗. (20) Then (1)𝑓 is well-defined if and only if for 1≤𝑖,𝑗≤𝑛, 𝑚_𝑗∣𝑚_𝑖𝑝_𝑖𝑗. (21)(2)If 𝑓 is well-defined, then(a)𝑓 is an ℤ-module homomorphism,(b)𝑓 is one to one if and only if for each 𝑠₁,𝑠₂,…,𝑠_𝑛∈ℤ,𝑚_𝑗∣∑^𝑚_𝑖=1𝑠_𝑖𝑝_𝑖𝑗 implies that 𝑚_𝑗∣𝑠_𝑗 for all 1≤𝑗≤𝑛, and (c)𝑓 is a ring homomorphism if and only if for 1≤𝑖,𝑗,𝑘≤𝑛, ^𝑛_𝛼,𝛽=1𝑝_𝑖𝛼𝑝_𝑗𝛽𝑤_𝛼𝛽𝑘≡_𝑚𝑘^𝑛_𝑙=1𝑧_𝑖𝑗𝑙𝑝_𝑙𝑘. (22)

Proof. (1) (⇒). Assume 𝑠∈𝑚_𝑖ℤ, where 1≤𝑖≤𝑛. Then by definition 𝑓 we have ^𝑛_𝑗=1𝑠𝑝_𝑖𝑗𝑏_𝑗=𝑓𝑠𝑎_𝑖=𝑓(0)=0. (23) It follows that 𝑠𝑝_𝑖𝑗𝑏_𝑗=0 for all 1≤𝑗≤𝑛. Therefore, 𝑚_𝑖𝑝_𝑖𝑗ℤ⊆𝑚_𝑗ℤ, for all 1≤𝑗≤𝑛 and so for 1≤𝑖,𝑗≤𝑛, 𝑚_𝑗∣𝑚_𝑖𝑝_𝑖𝑗. (24)
(1) (⇐). Suppose ∑^𝑛_𝑖=1𝑠_𝑖𝑎_𝑖=∑^𝑛_𝑖=1𝑡_𝑖𝑎_𝑖 where 𝑠_𝑖,𝑡_𝑖∈ℤ. Then 𝑠_𝑖−𝑡_𝑖∈𝑚_𝑖ℤ for all 𝑖 (1≤𝑖≤𝑛) and so (𝑠_𝑖−𝑡_𝑖)𝑝_𝑖𝑗∈𝑚_𝑖𝑝_𝑖𝑗ℤ. Now by our hypothesis 𝑚_𝑗∣(𝑠_𝑖−𝑡_𝑖)𝑝_𝑖𝑗thatis,𝑠_𝑖𝑝_𝑖𝑗𝑏_𝑗=𝑡_𝑖𝑝_𝑖𝑗𝑏_𝑗 for all 1≤𝑖,𝑗≤𝑛. Therefore, ^𝑛_𝑗=1^𝑛_𝑖=1𝑠_𝑖𝑝_𝑖𝑗𝑏_𝑗=^𝑛_𝑗=1^𝑛_𝑖=1𝑡_𝑖𝑝_𝑖𝑗𝑏_𝑗. (25) This means that 𝑓^𝑛_𝑖=1𝑠_𝑖𝑎_𝑖=𝑓^𝑛_𝑖=1𝑡_𝑖𝑎_𝑖 (26) and hence 𝑓 is well-defined.
(2) (a) Assume 𝑟,𝑠_𝑖,𝑡_𝑖∈ℤ, for 𝑖=1,…,𝑛. Clearly, 𝑓𝑟^𝑛_𝑖=1𝑠_𝑖𝑎_𝑖+^𝑛_𝑖=1𝑡_𝑖𝑎_𝑖=𝑓^𝑛_𝑖=1𝑟𝑠_𝑖+𝑡_𝑖𝑎_𝑖=^𝑛_𝑗=1^𝑛_𝑖=1𝑟𝑠_𝑖+𝑡_𝑖𝑝_𝑖𝑗𝑏_𝑗=^𝑛_𝑗=1^𝑛_𝑖=1𝑟𝑠_𝑖𝑝_𝑖𝑗𝑏_𝑗+^𝑛_𝑗=1^𝑛_𝑖=1𝑡_𝑖𝑝_𝑖𝑗𝑏_𝑗=𝑟𝑓^𝑛_𝑖=1𝑠_𝑖𝑎_𝑖+𝑓^𝑛_𝑖=1𝑡_𝑖𝑎_𝑖. (27) Thus 𝑓 is an ℤ-module homomorphism.
(2) (b) By definition 𝑓, (b) is clear.
(2) (c) By (2) (a), 𝑓 is always a group homomorphism. Thus it is sufficient to show that 𝑓(𝑎_𝑖𝑎_𝑗) = 𝑓(𝑎_𝑖)𝑓(𝑎_𝑗) for each 𝑖,𝑗(1≤𝑖,𝑗≤𝑛). However by multiplications of 𝑅₁, 𝑅₂ and definition of 𝑓 we have 𝑓𝑎_𝑖𝑎_𝑗=𝑓^𝑛_𝑙=1𝑤_𝑖𝑗𝑙𝑎_𝑙=^𝑛_𝑘=1^𝑛_𝑙=1𝑤_𝑖𝑗𝑙𝑝_𝑙𝑘𝑏_𝑘 (28) and also 𝑓𝑎_𝑖𝑓𝑎_𝑗=^𝑛_𝑠=1𝑝_𝑖𝑠𝑏_𝑠^𝑛_𝑡=1𝑝_𝑗𝑡𝑏_𝑡=^𝑛_𝑘=1^𝑛_𝑠,𝑡=1𝑝_𝑖𝑠𝑝_𝑗𝑡𝑧_𝑠𝑡𝑘𝑏_𝑘. (29) Thus 𝑓 is a ring homomorphism if and only if for all 1≤𝑖,𝑗,𝑘≤𝑛, ^𝑛_𝛼,𝛽=1𝑝_𝑖𝛼𝑝_𝑗𝛽𝑤_𝛼𝛽𝑘≡_𝑚𝑘^𝑛_𝑙=1𝑧_𝑖𝑗𝑙𝑝_𝑙𝑘. (30)

Corollary 5. Let 𝑛 be a positive integer and let 𝐴_𝑖=⟨𝑎_𝑖⟩ (1≤𝑖≤𝑛) be cyclic additive groups of orders 𝑚_𝑖, respectively. If 𝑅₁=𝑎₁,…,𝑎_𝑛∶𝑚_𝑖𝑎_𝑖=0,𝑎_𝑖𝑎_𝑗=^𝑛_𝑘=1𝑤_𝑖𝑗𝑘𝑎_𝑘,𝑅,𝑖,𝑗=1,…,𝑛₂=𝑎₁,…,𝑎_𝑛∶𝑚_𝑖𝑎_𝑖=0,𝑎_𝑖𝑎_𝑗=^𝑛_𝑘=1𝑧_𝑖𝑗𝑘𝑎_𝑘,𝑖,𝑗=1,…,𝑛 (31) are two presentations with suitable 𝑤_𝑖𝑗𝑘,𝑧_𝑖𝑗𝑘∈ℤ (1≤𝑖,𝑗,𝑘≤𝑛), then 𝑅₁≅𝑅₂ if and only if there exist 𝑝_𝑖𝑗∈ℤ (1≤𝑖,𝑗≤𝑛) such that(1)𝑚_𝑗∣𝑚_𝑖𝑝_𝑖𝑗, for 1≤𝑖,𝑗≤𝑛; (2)for all 𝑠_𝑖∈ℤ and 1≤𝑗≤𝑛, 𝑚_𝑗∣∑^𝑚_𝑖=1𝑠_𝑖𝑝_𝑖𝑗 implies that 𝑚_𝑗∣𝑠_𝑗 for all 1≤𝑗≤𝑛, and (3)∑^𝑛_𝛼,𝛽=1𝑝_𝑖𝛼𝑝_𝑗𝛽𝑤_𝛼𝛽𝑘≡_𝑚𝑘∑^𝑛_𝑙=1𝑧_𝑖𝑗𝑙𝑝_𝑙𝑘, for 1≤𝑖,𝑗,𝑘≤𝑛.

Proof. It is clear by Theorem 4.

3. Finite Rings of Order 𝑝⁶ with Characteristic 𝑝⁶ and 𝑝⁵

In this section, we first classify all finite rings of order 𝑝⁶ with characteristic 𝑝⁶. Then this classification is extended to the rings order 𝑝^𝑛 with characteristic 𝑝^𝑛.

Proposition 6. (i) Let 𝑅=⟨𝑎⟩ be an additive cyclic group of order 𝑝⁶ and 𝑤∈ℤ. Then 𝑅 = ⟨𝑎∶𝑝⁶𝑎=0,𝑎²=𝑤𝑎⟩ is always a ring.
(ii) Let 𝑅₁=⟨𝑎∶𝑝⁶𝑎=0,𝑎²=𝑤𝑎⟩ and 𝑅₂=⟨𝑎∶𝑝⁶𝑎=0,𝑎²=𝑧𝑎⟩ be two presentations with 𝑤,𝑧∈ℤ. Assume 𝑞∈ℤ and 𝑓 is the following map: 𝑓∶𝑅₁→𝑅₂,𝑓(𝑟𝑎)=𝑟𝑞𝑎. (32) Then (1)𝑓 is well-defined;(2)𝑓 is a ring homomorphism if and only if for 𝑤𝑞≡_𝑝6𝑧𝑞²;(3)𝑓 is one to one if and only if for each 𝑟∈ℤ,𝑝⁶∣𝑟𝑞 implies that 𝑝⁶∣𝑟.

Proof. (i) It is straightforward by relations (∗) and (∗∗) in Theorem 3.
(ii) The proof is obtained by simplification of the relations in Theorem 4.

Since the function introduced in preceding proposition is multiplication by some 𝑞∈ℤ, and is always well-defined, we have the following corollary.

Corollary 7. Let 𝑅₁=⟨𝑎∶𝑝⁶𝑎=0,𝑎²=𝑤𝑎⟩ and 𝑅₂=⟨𝑎∶𝑝⁶𝑎=0,𝑎²=𝑧𝑎⟩ be two presentations with 𝑤,𝑧∈ℤ. Then 𝑅₁≅𝑅₂ if and only if there exists 𝑞∈ℤ such that(1)𝑤𝑞≡_𝑝6𝑧𝑞²;(2)for each 𝑟∈ℤ,𝑝⁶∣𝑟𝑞 implies that 𝑝⁶∣𝑟.

Proof. It is clear by Proposition 6.

Theorem 8. For any prime number 𝑝, there are exactly seven rings of order 𝑝⁶ whose additive group is cyclic, only one of these rings have identity. However all of these are commutative.

Proof. Let 𝑅₁=⟨𝑎∶𝑝⁶𝑎=0,𝑎²=𝑤𝑎⟩ and 𝑅₂=⟨𝑎∶𝑝⁶𝑎=0,𝑎²=𝑧𝑎⟩ be two presentations with 𝑤,𝑧∈ℤ. The relation (2) in Corollary 7 implies that 𝑝 is relatively prime to 𝑞 and so by the relation (1) in Corollary 7, we deduce that 𝑤≡_𝑝6𝑧𝑞. Since the relation 𝑤≡_𝑝6𝑧𝑥 has a solution, gcd(𝑝⁶,𝑧)∣𝑤. Now we proceed by cases.
Case 1. gcd(𝑝,𝑤)=1, that is, 𝑤 is relatively prime to 𝑝. Since gcd(𝑝⁶,𝑧)∣𝑤, 𝑧 is also relatively prime to 𝑝. Thus 𝑅₁≅𝑅₂ where both 𝑤 and 𝑧 are relatively prime to 𝑝.
Case 2. gcd(𝑝²,𝑤)=𝑝. Then the relation (𝑝⁶,𝑧)∣𝑤 shows that either gcd(𝑧,𝑝)=1 or 𝑧∈{𝑝,2𝑝,3𝑝,…,(𝑝−1)𝑝}. But the relation 𝑤≡_𝑝6𝑧𝑞 follows that 𝑤≡_𝑝𝑧𝑞 and hence 𝑝∣𝑧𝑞 because gcd(𝑝²,𝑤)=𝑝. Since gcd(𝑝,𝑞)=1, 𝑝∣𝑧. Therefore we obtain that 𝑧∈{𝑝,2𝑝,3𝑝,…,(𝑝−1)𝑝} and so gcd(𝑝²,𝑧)=𝑝. Thus 𝑅₁≅𝑅₂ where both gcd(𝑝²,𝑤)=𝑝 and gcd(𝑝²,𝑧)=𝑝.
Case 3. gcd(𝑝³,𝑤)=𝑝². Then the relation gcd(𝑝⁶,𝑧)∣𝑤 shows that either gcd(𝑧,𝑝)=1 or 𝑧∈{𝑝,2𝑝,3𝑝,…,(𝑝−1)𝑝,𝑝²,2𝑝²,3𝑝²,…,(𝑝−1)𝑝²}. But the relation 𝑤≡_𝑝6𝑧𝑞 follows that 𝑤≡_𝑝2𝑧𝑞 and hence 𝑝²|𝑧𝑞 because gcd(𝑝²,𝑤)=𝑝. Since gcd(𝑝,𝑞)=1, 𝑝²∣𝑧. Therefore we obtain that 𝑧∈{𝑝²,2𝑝²,3𝑝²,…,(𝑝−1)𝑝²} and so gcd(𝑝³,𝑧)=𝑝². Thus 𝑅₁≅𝑅₂ where both gcd(𝑝³,𝑤)=𝑝² and gcd(𝑝³,𝑧)=𝑝².
Case 4. gcd(𝑝⁴,𝑤)=𝑝³. Then the relation gcd(𝑝⁶,𝑧)∣𝑤 shows that either gcd(𝑧,𝑝)=1 or 𝑧∈𝑝,2𝑝,3𝑝,…,𝑝−1𝑝,𝑝²,2𝑝²,3𝑝²𝑝,…,𝑝−1²,𝑝³,2𝑝³,3𝑝³,…,𝑝−1𝑝³. (33) But the relation 𝑤≡_𝑝6𝑧𝑞 follows that 𝑤≡_𝑝3𝑧𝑞 and hence 𝑝³∣𝑧𝑞 because gcd(𝑝⁴,𝑤)=𝑝³. Since gcd(𝑝,𝑞)=1, 𝑝³∣𝑧. Therefore we obtain that 𝑧∈{𝑝³,2𝑝³,3𝑝³,…,(𝑝−1)𝑝³} and so gcd(𝑝⁴,𝑧)=𝑝³. Thus 𝑅₁≅𝑅₂ where both gcd(𝑝⁴,𝑤)=𝑝³ and gcd(𝑝⁴,𝑧)=𝑝³.
Case 5. gcd(𝑝⁵,𝑤)=𝑝⁴. By a similar argument, we deduce 𝑅₁≅𝑅₂ where both gcd(𝑝⁵,𝑤)=𝑝⁴ and gcd(𝑝⁵,𝑧)=𝑝⁴.
Case 6. gcd(𝑝⁶,𝑤)=𝑝⁵. By a similar argument, we deduce 𝑅₁≅𝑅₂ where both gcd(𝑝⁶,𝑤)=𝑝⁵ and gcd(𝑝⁶,𝑧)=𝑝⁵.
Case 7. gcd(𝑝⁶,𝑤)=𝑝⁶. By a similar argument, we deduce 𝑅₁≅𝑅₂ where both gcd(𝑝⁶,𝑤)=𝑝⁶ and gcd(𝑝⁶,𝑧)=𝑝⁶.

An analysis similar to that in the proof of Theorem 8 shows that there are exactly 𝑛+1 rings of order 𝑝^𝑛 with characteristic 𝑝^𝑛. These rings are 𝑎∶𝑝^𝑛𝑎=0,𝑎²=𝑤𝑎, (34)

𝑤∈{1,𝑝,𝑝²,…,𝑝^𝑛}

We can obtain the following result by applying relations (∗) and (∗∗) in Theorem 3, for the rings of order 𝑝⁶ with characteristic 𝑝⁵. Also, this manner may be used for obtaining all finite rings of order 𝑝⁶ with characteristic 𝑝^𝑘, where 1≤𝑘≤4.

Corollary 9. Let 𝑅=𝐴₁⊕𝐴₂ where A₁ = <a₁> and A₂ = <a₂> be cyclic groups of order 𝑝 and 𝑝⁵, respectively. Assume 𝑤_𝑖𝑗𝑘∈ℤ where 1≤𝑖,𝑗,𝑘≤2. Then 𝑅=⟨𝑎₁,𝑎₂∶𝑝𝑎₁=𝑝⁵𝑎₂=0,𝑎_𝑖𝑎_𝑗=∑²_𝑘=1𝑤_𝑖𝑗𝑘𝑎_𝑘,𝑖,𝑗=1,2⟩ is a ring if and only if the following hold.(1)𝑝⁴∣𝑤₁₁₂,𝑝⁴∣𝑤₁₂₂ and 𝑝⁴∣𝑤₂₁₂; (2)𝑤₁₁₁𝑤₁₂₂≡_𝑝5𝑤₁₂₁𝑤₁₁₂;(3)𝑤₂₁₁𝑤₂₁₂≡_𝑝5𝑤₁₂₁𝑤₁₂₂≡_𝑝5𝑤₂₂₁𝑤₁₁₂;(4)𝑤₂₁₁𝑤₁₁₂≡_𝑝5𝑤₁₁₁𝑤₂₁₂+𝑤₁₁₂𝑤₂₂₂;(5)𝑤₂₁₁𝑤₁₂₂+𝑤₂₁₂𝑤₂₂₂≡_𝑝5𝑤₁₂₁𝑤₂₁₂+𝑤₁₂₂𝑤₂₂₂;(6)𝑤₂₂₁𝑤₁₂₂≡_𝑝5𝑤₂₁₂𝑤₂₂₁.

Application and simplification of the relations (1), (2), and (3) in Corollary 5, for the rings of order 𝑝⁶ with characteristic 𝑝⁵, give us the following result.

Corollary 10. Let 𝑅=𝑎₁,𝑎₂∶𝑝𝑎₁=𝑝⁵𝑎₂=0,𝑎_𝑖𝑎_𝑗=²_𝑘=1𝑤_𝑖𝑗𝑘𝑎_𝑘,𝑎,𝑖,𝑗=1,2𝑆=₁,𝑎₂∶𝑝𝑎₁=𝑝⁵𝑎₂=0,𝑎_𝑖𝑎_𝑗=²_𝑘=1𝑧_𝑖𝑗𝑘𝑎_𝑘,𝑖,𝑗=1,2 (35) be two rings with suitable integers 𝑤_𝑖𝑗𝑘 and 𝑧_𝑖𝑗𝑘 satisfied in Theorem 3, for 1≤𝑖,𝑗,𝑘≤2. Then 𝑅≅𝑆 if and only if there exist 𝑝_𝑖𝑗∈ℤ such that the following hold.(1)𝑝⁴∣𝑝₁₂; (2)If 𝑝∣𝑠₁𝑝₁₁+𝑠₂𝑝₂₁ and 𝑝⁵∣𝑠₁𝑝₁₂+𝑠₂𝑝₂₂, then 𝑝∣𝑠₁ and 𝑝⁵∣𝑠₂; (3)𝑝²₁₁≡_𝑝𝑧₁₁₁(𝑝₁₁+𝑝₂₁); (4)𝑝₁₁𝑝₂₁𝑤₁₁₁+𝑝₁₁𝑝₂₂𝑤₁₂₁≡_𝑝𝑧₁₂₁(𝑝₁₁+𝑝₂₁); (5)𝑝²₂₁𝑤₁₁₁+𝑝₂₂𝑝₂₁𝑤₂₁₁+𝑝²₂₂𝑤₂₂₁≡_𝑝𝑧₂₂₁(𝑝₁₁+𝑝₂₁); (6)𝑝²₁₁𝑤₁₁₂≡_𝑝5𝑧₁₁₂𝑝₂₂; (7)𝑝₂₁𝑝₁₁𝑤₁₁₂+𝑝₂₂𝑝₁₁𝑤₂₁₂+𝑝₂₂𝑝₁₂𝑤₂₂₂≡_𝑝5𝑧₂₁₂𝑝₂₂; (8)𝑝₁₁𝑝₂₁𝑤₁₁₂+𝑝₁₁𝑝₂₂𝑤₁₂₂+𝑝₁₂𝑝₂₂𝑤₂₂₂≡_𝑝5𝑧₁₂₂𝑝₂₂; (9)𝑝²₂₁𝑤₁₁₂+𝑝₂₂𝑝₂₁(𝑤₂₁₂+𝑤₁₂₂)+𝑝²₂₂𝑤₂₂₂≡_𝑝5𝑧₂₂₂(𝑝₁₂+𝑝₂₂).

In the following, one gives an algorithm for computing the finite rings whose abelian group is isomorphic to ℤ_𝑚1⊕ℤ_𝑚2⊕⋯⊕ℤ_𝑚𝑛. We represent the algorithm in the form of two sub-algorithms. In Algorithm 11, one lets (𝑚₁,𝑚₂,…,𝑚_𝑛) be the input and 𝐴 be the set of all sequences (𝑤_𝑖𝑗𝑘) that satisfy the condition 0≤𝑤_𝑖𝑗𝑘≤𝑚_𝑘−1, for any 1≤𝑖,𝑗,𝑘≤𝑛. Then one has the sequences (𝑤_𝑖𝑗𝑘) that satisfy the conditions (∗) and (∗∗) in Theorem 3 as the outputs of the algorithm. In fact, any sequence (𝑤_𝑖𝑗𝑘) that is an output of this algorithm shows a ring as the following: 𝑎₁,𝑎₂…,𝑎_𝑛∶𝑚_𝑖𝑎_𝑖=0,𝑎_𝑖𝑎_𝑗=^𝑛_𝑘=1𝑤_𝑖𝑗𝑘𝑎_𝑘,𝑖,𝑗=1,…𝑛. (36) In Algorithm 12, one represents the conditions required for the isomorphism of two rings based upon Theorem 4. In fact, one verifies the two sequences (𝑤_𝑖𝑗𝑘) and (𝑧_𝑖𝑗𝑘) obtained from Algorithm 11, under isomorphic conditions in Theorem 4. Therefore under Algorithms 11 and 12, one computes all the rings whose abelian group is isomorphic to ℤ_𝑚1⊕ℤ_𝑚2⊕⋯⊕ℤ_𝑚𝑛 and are not isomorphic. We have implemented Algorithms 11 and 12 in 𝐶⁺⁺ and presents a sample of its responses for the rings whose abelian group is isomorphic to ℤ₂⊕ℤ₃₂ in the following example.

Algorithm 11. (1) Input (𝑚₁,𝑚₂,…,𝑚_𝑛)
(2) Let 𝑁=∶{1,2,…,𝑛}
(3) For all (𝑖,𝑗)∈𝑁², let 𝜆_𝑖𝑗=∶gcd(𝑚_𝑖,𝑚_𝑗)
(4) Let 𝐴=∶{𝑤=(𝑤_𝑖𝑗𝑘)_{(𝑖,𝑗,𝑘)∈𝑁}³; for all (𝑖,𝑗,𝑘)∈𝑁³,0≤𝑤_𝑖𝑗𝑘≤𝑚_𝑘−1}
(5) 𝐵=∶{}
(6) If 𝐴=𝐵, then go to end
(7) Choose 𝑤=(𝑤_𝑖𝑗𝑘)_{(𝑖,𝑗,𝑘)∈𝑁}³∈𝐴⧵𝐵
(8) Let 𝐵=∶𝐵∪{𝑤}
(9) If there is (𝑖,𝑗,𝑘)∈𝑁³ such that 𝜆_𝑖𝑗𝑤_𝑖𝑗𝑘≇_𝑚𝑘0, then go to step 6
(10) If there is (𝑖,𝑗,𝑘,𝑙)∈𝑁⁴ such that ∑^𝑛_𝛼=1𝑤_𝑖𝑗𝛼𝑤_𝛼𝑙𝑘≇_𝑚𝑘∑^𝑛_𝛼=1𝑤_𝑗𝑙𝛼𝑤_𝑖𝛼𝑘, then go to step 6
(11) Print 𝑤=(𝑤_𝑖𝑗𝑘)_{(𝑖,𝑗,𝑘)∈𝑁}³
(12) End

Algorithm 12. (1) Input (𝑚₁,𝑚₂,…,𝑚_𝑛)
(2) Let 𝑁=∶{1,2,…,𝑛}
(3) For all (𝑖,𝑗)∈𝑁², let 𝜆_𝑖𝑗=∶gcd(𝑚_𝑖,𝑚_𝑗)
(4) Input 𝑤=(𝑤_𝑖𝑗𝑘)_{(𝑖,𝑗,𝑘)∈𝑁}³ and 𝑧=(𝑧_𝑖𝑗𝑘)_{(𝑖,𝑗,𝑘)∈𝑁}³ satisfied in Algorithm 11
(5) Let 𝐴=∶{𝑃=(𝑝_𝑖𝑗)_{(𝑖,𝑗)∈𝑁}²; for all (𝑖,𝑗)∈𝑁²,0≤𝑝_𝑖𝑗≤𝑚_𝑗−1}
(6) 𝐵=∶{}
(7) If 𝐴=𝐵, then go to end
(8) Choose 𝑃=(𝑝_𝑖𝑗)_{(𝑖,𝑗)∈𝑁}²∈𝐴⧵𝐵
(9) Let 𝐵=∶𝐵∪{𝑃}
(10) If there is (𝑖,𝑗)∈𝑁² such that 𝑚_𝑖𝑃_𝑖𝑗≇_𝑚𝑗0, then go to step 7
(11) Let 𝐶=∶{}
(12) Let 𝐷=∶{𝑟=(𝑟_𝑗)_𝑗∈𝑁; for all 𝑗∈𝑁,0≤𝑟_𝑗≤𝑚_𝑗−1,∑^𝑛_𝑖=1𝑟_𝑖𝑝_𝑖𝑗≅_𝑚𝑗0}
(13) Choose 𝑟∈𝐷
(14) Let 𝐶=∶𝐶∪{𝑟}
(15) If there is 𝑖∈𝑁 such that 𝑟_𝑖≇_𝑚𝑖0, then go to step 7
(16) If there is (𝑖,𝑗,𝑘)∈𝑁³ such that ∑^𝑛_𝛼,𝛽=1𝑝_𝑖𝛼𝑝_𝑗𝛽𝑤_𝛼𝛽𝑘≇_𝑚𝑘∑^𝑛_𝑙=1𝑧_𝑖𝑗𝑙𝑝_𝑙𝑘, then go to step 7
(17) Print two rings 𝑅 and 𝑆 with representations 𝑤=(𝑤_𝑖𝑗𝑘)_{(𝑖,𝑗,𝑘)∈𝑁}³ and 𝑧=(𝑧_𝑖𝑗𝑘)_{(𝑖,𝑗,𝑘)∈𝑁}³, respectively, are isomorphic
(18) End

Example 13. We compute the presentations of all finite rings of order 2⁶, whose abelian group is isomorphic to ℤ₂⊕ℤ₃₂. Let 𝑅 be a such ring. Then we write the following presentation for 𝑅=⟨𝑎₁,𝑎₂∶2𝑎₁=2⁵𝑎₂=0,𝑎_𝑖𝑎_𝑗=𝑤_𝑖𝑗1𝑎₁+𝑤_𝑖𝑗2𝑎₂,𝑖,𝑗=1,2⟩: 𝑤𝑅=¹¹¹𝑤₁₁₂𝑤₁₂₁𝑤₁₂₂𝑤₂₁₁𝑤₂₁₂𝑤₂₂₁𝑤₂₂₂. (37)
The outputs of these algorithms for the rings of order 2⁶ whose abelian group is ℤ₂⊕ℤ₃₂ are listed below. 00000000,00000001,00000002,00000004,00000008,0000000,0000000,0000002,0000004,0000008,0000000,0000002,0000004,0000008,000000,000002,000004,000008,00000,0000000,0000002,0000004,0000008,000000,000000,000002,000004,000008,00000,00000010,00000012,00000014,00000018,000010,000012,000014,000018,00100001,00001001,00101001,0101001,10000000,10000001,10000002,10000004,10000008,1000000.1616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161616 (38)
Thus there exist 47 rings of order 2⁶ with characteristic 2⁵.

Acknowledgment

This research was supported by Islamic Azad University, Khorramabad Branch.