Weakly Semicommutative Rings and Strongly Regular Rings
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A ring R is called weakly semicommutative ring if for any a,b ∈ R* = R ＼{0} with ab=0, there exists n ≥ 1 such that either an ≠ 0 and anRb = 0 or bn ≠ 0 and aRbn = 0. In this paper, many properties of weakly semicommutative rings are introduced, some known results are extended. Especially, we show that a ring R is a strongly regular ring if and only if R is a left SF-ring and weakly semicommutative ring.
Abstract
A ring R is called weakly semicommutative ring if for any a,b ∈ R* = R ＼{0} with ab=0, there exists n ≥ 1 such that either an ≠ 0 and anRb = 0 or bn ≠ 0 and aRbn = 0. In this paper, many properties of weakly semicommutative rings are introduced, some known results are extended. Especially, we show that a ring R is a strongly regular ring if and only if R is a left SF-ring and weakly semicommutative ring.
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