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|Paper IPM / M / 11361||
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 Ri (1 ≤ i ≤ m) for some m ≥ 1. In this paper, we investigate the relationship between the prime factorizations |Z(R)|=p1k1…pnkn and the summands Ri. It is shown that for each i, |Z(Ri)|=pjtj for some 1 ≤ j ≤ n and 0 ≤ tj ≤ kj. In particular, rings R with |Z(R)|=pk where 1 ≤ k ≤ 7, are characterized. Moreover, the structure and classification up to isomorphism of all
commutative rings R with |Z(R)|=p1k1…pnkn,
where n ∈ \BbbN, pi,s are distinct prime numbers, 1 ≤ ki ≤ 3 and nonlocal commutative rings R with |Z(R)|=pk where k=4 or 5, are determined.
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