Semiderivations and Commutativity in Prime Rings
1988 ◽
Vol 31
(4)
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pp. 500-508
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AbstractA semiderivation of a ring R is an additive mapping f:R → R together with a function g:R → R such that f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y) and f(g(x) ) = g(f(x)) for all x, y ∊ R. Motivating examples are derivations and mappings of the form x → x — g(x), g a ring endomorphism. A semiderivation f of R is centralizing on an ideal U if [f(u), u] is central for all u ∊ U. For R prime of char. ≠2, U a nonzero ideal of R, and 0 ≠ f a semiderivation of R we prove: (1) if f is centralizing on U then either R is commutative or f is essentially one of the motivating examples, (2) if [f(U), f(U) ] is central then R is commutative.
2012 ◽
Vol 31
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pp. 65-70
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2014 ◽
Vol 33
(2)
◽
pp. 179-186
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1984 ◽
Vol 27
(1)
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pp. 122-126
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2018 ◽
Vol 11
(1)
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pp. 79
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