On Clean Rings |
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| Authors: | Hongbo Zhang Victor Camillo |
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| Institution: | 1. School of Physics and Mathematics, Changzhou University, Changzhou, Jiangsu, Chinahbzhang1212@aliyun.com;3. Department of Mathematics, The University of Iowa, Iowa City, Iowa, USA |
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| Abstract: | A ring R is called clean if every element of R is the sum of an idempotent and a unit. Let M be a R-module. It is obtained in this article that the endomorphism ring End(M) is clean if and only if, whenever A = M′ ⊕ B = A1 ⊕ A2 with M′ ? M, there is a decomposition M′ =M1 ⊕ M2 such that A = M′ ⊕ A1 ∩ (M1 ⊕ B)] ⊕ A2 ∩ (M2 ⊕ B)]. Then unit-regular endomorphism rings are also described by direct decompositions. |
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| Keywords: | Clean rings Endomorphism ring Unit-regular rings |
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