# Rings with zero intersection property on annihilators: Zip rings.

• Volume: 33, Issue: 2, page 329-338
• ISSN: 0214-1493

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## Abstract

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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.

## How to cite

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Faith, Carl. "Rings with zero intersection property on annihilators: Zip rings.." Publicacions Matemàtiques 33.2 (1989): 329-338. <http://eudml.org/doc/41099>.

@article{Faith1989,
abstract = {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. },
author = {Faith, Carl},
journal = {Publicacions Matemàtiques},
keywords = {Anillos; right zip rings; right annihilators; finitely generated faithful modules; injective right modules; left and right self-injective rings; pseudo-Frobenius rings; local quotient rings; FPF zip rings},
language = {eng},
number = {2},
pages = {329-338},
title = {Rings with zero intersection property on annihilators: Zip rings.},
url = {http://eudml.org/doc/41099},
volume = {33},
year = {1989},
}

TY - JOUR
AU - Faith, Carl
TI - Rings with zero intersection property on annihilators: Zip rings.
JO - Publicacions Matemàtiques
PY - 1989
VL - 33
IS - 2
SP - 329
EP - 338
AB - 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.
LA - eng
KW - Anillos; right zip rings; right annihilators; finitely generated faithful modules; injective right modules; left and right self-injective rings; pseudo-Frobenius rings; local quotient rings; FPF zip rings
UR - http://eudml.org/doc/41099
ER -

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