| Resum: |
Zelmanowitz [12] introduced the concept of ring, which we call right zip rings, with the defining properties below, whidt are equivalent : (ZIP 1) If the right annihilator X ┴ of a subset X of R is zero,then X,┴ = 0 for a finite subset X, C X. (ZIP 2) If L is a left ideal and if L┴ = 0, then L,┴ = 0 for a finitely generated left ideal L, C L. In [12], Zelmanowitz noted that any ring R satisfying the d. c. c. on annihilator right ideals (= dcc ┴) is a right zip ring, and hence, so is any subring of R. He also showed by example that there exist zip rings which do not have dcc ┴. In §1 of this paper, we characterize a right zip by the property that every injective right module E is divisible by every left ideal L such that L┴ = 0 . Thus, E = EL. (It suffices for this to hold for the injective hull of R. ). In §2 we show that a left and right self-injective ring R is zip iff R is pseudo-Frobenius (= PF) . We then apply this result to show that a serniprime commutative ring R is zip iff R is Goldie. In §3 we continue the study of commutative zip rings. |
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