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| Paper IPM / M / 18311 |
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| Abstract: | |
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A ring R is called clean (resp., semi-clean) if each element of R can be written as the sum of a unit and an idempotent (resp., a periodic element). In this paper, we seek the necessary and sufficient conditions under which, for a ring R and a group G with additional
conditions, the group ring RG is clean or semi-clean. As a remarkable result, let R be an abelian clean ring, a reduced ring, or a commutative ring, and let G be a locally nilpotent group, we show that, if RG is semi-clean, then G is locally finite. Also, we show that if R is a semi-local ring whose Jacobson radical is locally nilpotent and G is a locally finite
group, then RG is clean. These results generalize some earlier results in the literature.
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