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Paper   IPM / M / 11361
School of Mathematics
  Title:   On the structure of commutative rings with pk1...1pknn(1 ≤ ki ≤ 7) zero-divisors
  Author(s):  M. Behboodi (R. Beyranvand)
  Status:   Published
  Journal: European Journal of Pure and Applied Mathematics
  Vol.:  3
  Year:  2010
  Pages:   303-316
  Supported by:  IPM
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 ≤ im) for some m ≥ 1. In this paper, we investigate the relationship between the prime factorizations |Z(R)|=p1k1pnkn and the summands Ri. It is shown that for each i, |Z(Ri)|=pjtj for some 1 ≤ jn and 0 ≤ tjkj. 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)|=p1k1pnkn, 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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