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# 17 - M. Behboodi (with R. Beyranvand), On the Structure of Commutative rings with p^k_1…p^k_n (1\leq k_i\leq 7) zero divisors, Euro. J. Pure Appl. Math, 3 (2) (2010), 303-316.

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 R_i (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(R_i)| = p^j^t j 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 all commutative rings R with |Z(R)| = p1^k1 . . . pn^kn , where n ∈ N, p, is are distinct prime numbers, 1 ≤ k_i ≤ 3 and nonlocal commutative rings R with |Z(R)| = pk where k = 4 or 5, are determined.