T-dual Rickart modules

Document Type : Research Paper

Authors

1 Department of Mathematics, University‎ ‎of Guilan‎, ‎P.O. Box 1914, Rasht‎, ‎Iran.

2 Department of‎ ‎Mathematics, University‎ ‎of Guilan‎, ‎P.O. Box 1914, Rasht‎, ‎Iran.

3 Department of Mathematics, ‎University‎ ‎of Guilan‎, ‎P.O. Box 1914, Rasht‎, ‎Iran.

Abstract

We introduce the notions of T-dual Rickart and strongly T-dual Rickart modules. We provide several characterizations and investigate properties of each of these concepts. It is shown that every free (resp. finitely generated free) $R$-module is T-dual Rickart if and only if $\overline{Z}^2(R)$ is a  direct summand of $R$ and End$(\overline{Z}^2(R))$ is a semisimple (resp. regular) ring. It is shown that, while a direct summand of a (strongly) T-dual Rickart module inherits the property, direct sums of T-dual Rickart modules do not. Moreover, when a direct sum of T-dual Rickart modules is T-dual Rickart, is included. Examples
illustrating the results are presented.

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References

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History

  • Receive Date: 24 September 2013
  • Revise Date: 15 March 2015
  • Accept Date: 15 March 2015

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