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Weakly Projective and Weakly Injective Modules

Published online by Cambridge University Press:  20 November 2018

S. K. Jain ,
S. R. López-Permouth ,
K. Oshiro  and
M. A. Saleh
S. K. Jain
Affiliation:
Ohio University Athens, Ohio 45701 U.S.A.
S. R. López-Permouth
Affiliation:
Ohio University Athens, Ohio 45701 U.S.A.
K. Oshiro
Affiliation:
Yamaguchi University Yoshido, Yamaguchi Japan
M. A. Saleh
Affiliation:
BirZeit University, BirZeit West Bank
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Abstract

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A module M is said to be weakly N-projective if it has a projective cover π: P(M) ↠M and for each homomorphism : P(M) → N there exists an epimorphism σ:P(M) ↠M such that (kerσ) = 0, equivalently there exists a homomorphism :M ↠N such that σ= . A module M is said to be weakly projective if it is weakly N-projective for all finitely generated modules N. Weakly N-injective and weakly injective modules are defined dually. In this paper we study rings over which every weakly injective right R-module is weakly projective. We also study those rings over which every weakly projective right module is weakly injective. Among other results, we show that for a ring R the following conditions are equivalent:

(1) R is a left perfect and every weakly projective right R-module is weakly injective.

(2) R is a direct sum of matrix rings over local QF-rings.

(3) R is a QF-ring such that for any indecomposable projective right module eR and for any right ideal I, soc(eR/eI) = (eR/eJ)n for some positive integer n.

(4) R is right artinian ring and every weakly injective right R-module is weakly projective.

(5) Every weakly projective right R-module is weakly injective and every weakly injective right R-module is weakly projective.

Keywords

16D4016D5016L3016L6016P20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 5 , 01 October 1994 , pp. 971 - 981
Copyright
Copyright © Canadian Mathematical Society 1994

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