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Abstract
Abelian groups whose endomorphism rings are von Neumann regular have been extensively investigated in the literature. In this paper, we study modules whose endomorphism rings are von Neumann regular, which we call endoregular modules. We provide characterizations of endoregular modules and investigate their properties. Some classes of rings R are characterized in terms of endoregular R-modules. It is shown that a direct summand of an endoregular module inherits the property, while a direct sum of endoregular modules does not. Necessary and sufficient conditions for a finite direct sum of endoregular modules to be an endoregular module are provided. As a special case, modules whose endomorphism rings are semisimple artinian are characterized. We provide a precise description of an indecomposable endoregular module over an arbitrary commutative ring. A structure theorem for extending an endoregular abelian group is also provided.
ACKNOWLEDGMENTS
The authors express their gratitude to the referee for a prompt and thorough report. Referee's suggestions have significantly improved the presentation of the paper. The authors are very thankful to the Math Research Institute, Columbus, the Ohio State University, Columbus and OSU-Lima, for the support of this research work. We also express sincere thanks to Professor Jae K. Park and Professor X. Zhang for several helpful discussions.
Notes
Communicated by T. Albu.