Let R be a ring and let g be an endomorphism of R. The additive mapping d: R -> R is called a Jordan semiderivation of R, associated with g, if d(x(2)) = d(x)x + g(x)d(x) = d(x)g(x) + xd(x) and d(g(x)) = g(d(x)) for all x is an element of R. The additive mapping F: R -> R is called a generalized Jordan semiderivation of R, related to the Jordan semiderivation d and endomorphism g, if F(x(2)) = F(x)x + g(x)d(x) = F(x)g(x) + xd(x) and F(g(x)) = g(F(x)) for all x is an element of R. In this paper we prove that if R is a prime ring of characteristic different from 2, g an endomorphism of R, d a Jordan semiderivation associated with g, F a generalized Jordan semiderivation associated with d and g, then F is a generalized semiderivation of R and d is a semiderivation of R. Moreover, if R is commutative, then F = d.
Generalized Jordan Semiderivations in Prime Rings
DE FILIPPIS, Vincenzo;
2015-01-01
Abstract
Let R be a ring and let g be an endomorphism of R. The additive mapping d: R -> R is called a Jordan semiderivation of R, associated with g, if d(x(2)) = d(x)x + g(x)d(x) = d(x)g(x) + xd(x) and d(g(x)) = g(d(x)) for all x is an element of R. The additive mapping F: R -> R is called a generalized Jordan semiderivation of R, related to the Jordan semiderivation d and endomorphism g, if F(x(2)) = F(x)x + g(x)d(x) = F(x)g(x) + xd(x) and F(g(x)) = g(F(x)) for all x is an element of R. In this paper we prove that if R is a prime ring of characteristic different from 2, g an endomorphism of R, d a Jordan semiderivation associated with g, F a generalized Jordan semiderivation associated with d and g, then F is a generalized semiderivation of R and d is a semiderivation of R. Moreover, if R is commutative, then F = d.Pubblicazioni consigliate
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