Dual Rickart Modules

Abstract

Rickart property for modules has been studied recently. In this article, we introduce and study the notion of dual Rickart modules. A number of characterizations of dual Rickart modules are provided. It is shown that the class of rings R for which every right R-module is dual Rickart is precisely that of semisimple artinian rings, the class of rings R for which every finitely generated free R-module is dual Rickart is exactly that of von Neumann regular rings, while the class of rings R for which every injective R-module is dual Rickart is precisely that of right hereditary ones. We show that the endomorphism ring of a dual Rickart module is always left Rickart and obtain conditions for the converse to hold true. We prove that a dual Rickart module with no infinite set of nonzero orthogonal idempotents in its endomorphism ring is a dual Baer module. A structure theorem for a finitely generated dual Rickart module over a commutative noetherian ring is provided. It is shown that, while a direct summand of a dual Rickart module inherits the property, direct sums of dual Rickart modules do not. We introduce the notion of relative dual Rickart property and show that if M _i is M _j -projective for all i > j ∈ ℐ = {1, 2,…, n} then is a dual Rickart module if and only if M _i is M _j -d-Rickart for all i, j ∈ ℐ. Other instances of when a direct sum of dual Rickart modules is dual Rickart, are included. Examples which delineate the concepts and results are provided.

Key Words:

• Dual Baer modules
• Endomorphism rings
• Idempotents and annihilator
• Rickart and Baer rings and modules
• von Neumann regular rings

2000 Mathematics Subject Classification:

• Primary 16D10, 16D70
• Secondary 16D40, 16S50, 16D80

ACKNOWLEDGMENTS

The authors are very thankful to the referee for his/her nice comments about this work. We also very much appreciate a prompt and thorough report from the referee on the article. We express our thanks to Math Research Institute, Ohio State University, Columbus and OSU-Lima for the support of this research work.

Notes

Communicated by T. Albu.

S. Tariq Rizvi's current address is Department of Mathematics, College of Science, Al-Imam University, Riyadh 11623, KSA.