Direct sums of Rickart modules

JOURNAL OF ALGEBRA

卷:353

Article

Lee, Gangyong¹  Rizvi, S. Tariq²  Roman, Cosmin S.² 

[1] Ohio State Univ, Dept Math, Columbus, OH 43210 USA

[2] Ohio State Univ, Dept Math, Lima, OH 45804 USA

关键词: Rickart (p.p.) rings and modules;    Baer rings and modules;    Idempotents;    Annihilators;    Endomorphisms;    Direct sums of modules;    Right (semi)hereditary;    Free and projective modules;   

DOI  :  10.1016/j.jalgebra.2011.12.003

来源: Elsevier

【 摘 要 】

The notion of Rickart modules was defined recently. It has been shown that a direct sum of Rickart modules is not a Rickart module, in general. In this paper we investigate the question: When are the direct sums of Rickart modules, also Rickart? We show that if M-i is M-j-injective for all i < j is an element of I = {1, 2,..., n} then circle plus(n)(i=1), M-i, is a Rickart module if and only if M-i is M-j-Rickart for all i, j is an element of I. As a consequence we obtain that for a nonsingular extending module M, E(M) circle plus M is always a Rickart module. Other characterizations for direct sums to be Rickart under certain assumptions are provided. We also investigate when certain classes of free modules over a ring R, are Rickart. It is shown that every finitely generated free R-module is Rickart precisely when R is a right semihereditary ring. As an application, we show that a commutative domain R is Prufer if and only if the free R-module R-(2) is Rickart. We exhibit an example of a module M for which M-(2) is Rickart but M-(3) is not so. Further, von Neumann regular rings are characterized in terms of Rickart modules. It is shown that the class of rings R for which every finitely cogenerated right R-module is Rickart, is precisely that of right V-rings. Examples which delineate the concepts and the results are provided. Published by Elsevier Inc.

【 授权许可】

Free