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

## Keywords:

Finite ring, Zero-divisor, Local rings## Abstract

Let $R$ be a finite commutative ringÂ with identity and $Z(R)$ denote the set of all zero-divisors of Â $R$.Â Note thatÂ $R$Â is uniquely expressible as a direct sum of local rings $R_i$ ($1\leq i\leq m$) for some $m\geq 1$. 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 $1\leq j\leq n$ and $0\leq t_j\leq k_j$.Â In particular, rings $R$ with $|Z(R)|=p^k$ where $1\leq k\leq 7$, are characterized. Moreover, the structure and classification up to isomorphism of allcommutative rings $R$ with $|Z(R)|={p_1}^{k_1}\ldots {p_n}^{k_n}$,

where $n\in \Bbb{N}$, $p_i^,s$ are distinct prime numbers, $1\leq k_i\leq 3$ and nonlocal commutative rings $R$ with $|Z(R)|=p^k$ whereÂ $ k=4$ or $5$, are determined.

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## Published

2010-04-09

## Issue

## Section

Algebra

## License

Upon acceptance of an article by the journal, the author(s) accept(s) the transfer of copyright of the article to *European Journal of Pure and Applied Mathematics.*

*European Journal of Pure and Applied Mathematics will be Copyright Holder.*

## How to Cite

*European Journal of Pure and Applied Mathematics*,

*3*(2), 303-316. https://www.ejpam.com/index.php/ejpam/article/view/628