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

Journal Name:

  • European Journal of Pure and Applied Mathematics

Publication Year:

  • 2010

Key Words:

Author NameUniversity of Author

AMS Codes:

Abstract (2. Language): 
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)| = p1 k1 · · · pn kn and the summands Ri . It is shown that for each i, |Z(Ri )| = pj t j for some 1 ≤ j ≤ n and 0 ≤ t j ≤ kj . In particular, rings R with |Z(R)| = pk where 1 ≤ k ≤ 7, are characterized. Moreover, the structure and classification up to isomorphism all commutative rings R with |Z(R)| = p1 k1 . . . pn kn , where n ∈ N, p, is 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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  • 2
303-316
  • English

REFERENCES

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