On rings with associated elements |
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Authors: | M Tamer Ko?an Truong Cong Quynh Serap ?ahinkaya |
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Institution: | 1. Department of Mathematics, Gebze Technical University, Gebze, Kocaeli, Turkeymtkosan@gyte.edu.tr tkosan@gmail.com;3. Department of Mathematics, Danang University, DaNang, Vietnam;4. Department of Mathematics, Gebze Technical University, Gebze, Kocaeli, Turkey |
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Abstract: | A principal right ideal of a ring is called uniquely generated if any two elements of the ring that generate the same principal right ideal must be right associated (i.e., if for all a,b in a ring R, aR = bR implies a = bu for some unit u of R). In the present paper, we study “uniquely generated modules” as a module theoretic version of “uniquely generated ideals,” and we obtain a characterization of a unit-regular endomorphism ring of a module in terms of certain uniquely generated submodules of the module among some other results: End(M) is unit-regular if and only if End(M) is regular and all M-cyclic submodules of a right R-module M are uniquely generated. We also consider the questions of when an arbitrary element of a ring is associated to an element with a certain property. For example, we consider this question for the ring Rx;σ]∕(xn+1), where R is a strongly regular ring with an endomorphism σ be an endomorphism of R. |
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Keywords: | Associated elements regular ring skew polynomial ring strongly regular ring uniquely generated module unit-regular ring |
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