On strong reversible rings and their extensions
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P. M. Cohn called a ring R reversible if whenever ab=0,then ba=0 for a, b ∈ R. In this paper, we study an extension of a reversible ring with its endomorphism. An endomorphism α of a ring R is called strong right (resp.,left) reversible if whenever aα(b)=0 (resp.,aα(a)b=0) for a, b ∈ R, ba=0. A ring R is called strong right (resp., left)α-reversible if there exists a strong right (resp., left)reversible endomorphism α of R, and the ring R is called strong α-reversible if R is both strong left and right α-reversible. We investigate characterizations of strong α-reversible rings and their related properties including extensions. In particular, we show that every semiprime and strong α-reversible ring is α-rigid and that for an α-skew Armendariz ring R, the ring R is reversible and strong α-reversible if and only if the skew polynomial ring R[x;α] of R is reversible.
Abstract
P. M. Cohn called a ring R reversible if whenever ab=0,then ba=0 for a, b ∈ R. In this paper, we study an extension of a reversible ring with its endomorphism. An endomorphism α of a ring R is called strong right (resp.,left) reversible if whenever aα(b)=0 (resp.,aα(a)b=0) for a, b ∈ R, ba=0. A ring R is called strong right (resp., left)α-reversible if there exists a strong right (resp., left)reversible endomorphism α of R, and the ring R is called strong α-reversible if R is both strong left and right α-reversible. We investigate characterizations of strong α-reversible rings and their related properties including extensions. In particular, we show that every semiprime and strong α-reversible ring is α-rigid and that for an α-skew Armendariz ring R, the ring R is reversible and strong α-reversible if and only if the skew polynomial ring R[x;α] of R is reversible.
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