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Paper IPM / M / 11361  


Abstract:  
Let R be a finite commutative ring with identity and Z(R) denote the set of all zerodivisors of R. Note that R is uniquely expressible as a direct sum of local rings R_{i} (1 ≤ i ≤ m) for some m ≥ 1. In this paper, we investigate the relationship between the prime factorizations Z(R)=p_{1}^{k1}…p_{n}^{kn} and the summands R_{i}. It is shown that for each i, Z(R_{i})=p_{j}^{tj} for some 1 ≤ j ≤ n and 0 ≤ t_{j} ≤ k_{j}. In particular, rings R with Z(R)=p^{k} where 1 ≤ k ≤ 7, are characterized. Moreover, the structure and classification up to isomorphism of all
commutative rings R with Z(R)=p_{1}^{k1}…p_{n}^{kn},
where n ∈ \BbbN, p_{i}^{,}s are distinct prime numbers, 1 ≤ k_{i} ≤ 3 and nonlocal commutative rings R with Z(R)=p^{k} where k=4 or 5, are determined.
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