On the Structure of Commutative Rings with ${\bf {p_1}^{k_1}\cdots {p_n}^{k_n} Zero-Divisors

Abstract

Let $R$ be a finite commutative ring  with identity and $Z(R)$ denote the set of all zero-divisors of  $R$.  Note that  $R$  is uniquely expressible as a direct sum of local rings $R_i$ ($1\le i\le m$) for some $m\ge 1$. In this paper, we investigate the relationship between the prime factorizations $|Z(R)|={p_1}^{k_1}\cdots {p_n}^{k_n}$ and the summands $R_i$. It is shown that for each $i$, $|Z(R_i)|={p_j}^{t_j}$ for some $1\le j\le n$ and $0\le t_j\le k_j$.  In particular, rings $R$ with $|Z(R)|=p^k$ where $1\le k\le 7$, are characterized. Moreover, the structure and classification up to isomorphism of all
commutative rings $R$ with $|Z(R)|={p_1}^{k_1}\dots {p_n}^{k_n}$,
where $n\in \mathbb{N}$, $p_i^{,}s$ are distinct prime numbers, $1\le k_i\le 3$ and nonlocal commutative rings $R$ with $|Z(R)|=p^k$ where  $k=4$ or $5$, are determined.