Rings with zero intersection property on annihilators: Zip rings.

Faith, Carl

Publicacions Matemàtiques, Tome 33 (1989), p. 329-338 / Harvested from Biblioteca Digital de Matemáticas

Résumé

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 ⊆ X.

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

Détail

Publié le : 1989-01-01

MR MR1030970 | Zbl 0702.16015

DMLE-ID : 3653