On FC-Purity and I-Purity of Modules and Köthe Rings

Abstract

In this article, several characterizations of certain classes of rings via FC-purity and I-purity are considered. Among others results, it is shown that every I-pure injective left R-module is projective if and only if every FC-pure projective left R-module is injective, if and only if, R is a semisimple ring. In particular, the structures of FC-pure projective and I-pure projective modules over a left Artinian ring are completely described. Also, it is shown that every left R-module is FC-pure projective if and only if every indecomposable left R-module is a finitely presented cyclic R-module, if and only if, R is a left Köthe ring. Finally, we introduce FC-pure flatness and I-pure flatness of modules and several characterizations of these notions are given. In particular, we show that a commutative ring R is quasi-Frobenius if and only if R is an Artinian ring and I-pure injective, if and only if, R is an Artinian ring and the injective envelope E(R) is an FC-pure projective R-module.

Key Words:

• FC-pure projective
• FC-pure injective
• FC-pure flat
• I-pure projective
• I-pure injective
• I-pure flat
• Köthe ring

2010 Mathematics Subject Classification:

• 16D40
• 16D50
• 16L60
• 13C10
• 13C11

ACKNOWLEDGMENTS

The authors owe a great debt to the referee who has carefully read an earlier version of this article and made significant suggestions for improvement. We would like to express our deep appreciation for the referee's work.

The research of the first author was in part supported by a grant from IPM (No. 91130413).

Notes

Communicated by T. Albu.