Dual Rickart Modules

• Published in: Communications in algebra Vol. 39; no. 11; pp. 4036 - 4058
• Main Authors: Lee, Gangyong, Rizvi, S. Tariq, Roman, Cosmin S.
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 01-11-2011
• Subjects: Dual Baer modules; Endomorphism rings; Idempotents and annihilator; Primary 16D10, 16D70; Rickart and Baer rings and modules; Secondary 16D40, 16S50, 16D80; von Neumann regular rings
• ISSN: 0092-7872; 1532-4125
• DOI: 10.1080/00927872.2010.515639

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.