Rings with zero intersection property on annihilators: zip rings
- Autores: Carl Faith
- Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 33, Nº 2, 1989, págs. 329-338
- Idioma: inglés
- DOI: 10.5565/publmat_33289_09
- Títulos paralelos:
- Anillos con propiedad de intersección cero sobre aniquiladores: anillos zip
- Enlaces
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Resumen
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.
