Resumen de Rings with zero intersection property on annihilators: zip rings

Carl Faith

• Zelmanowitz [12] introduced the concept of ring, which we call right zip rings, with the defining properties below, which are equivalent:

  (ZIP 1) If the right anihilator X^ of a subset X of R is zero, then X1^ = 0 for a finite subset X1 Í X.

  (ZIP 2) If L is a left ideal and if L^ = 0, then L1^ = 0 for a finitely generated left ideal L1 Í L.

  In [12], Zelmanowitz noted that any ring R satisfying the d.c.c. on anihilator 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 paragraph 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 paragraph 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 semiprime commutative ring R is zip iff R is Goldie.

  In paragraph 3 we continue the study of commutative zip rings.