ON LEFT α-MULTIPLIERS AND COMMUTATIVITY OF SEMIPRIME RINGS

Title & Authors
ON LEFT α-MULTIPLIERS AND COMMUTATIVITY OF SEMIPRIME RINGS
Ali, Shakir; Huang, Shuliang;

Abstract
Let R be a ring, and $\small{{\alpha}}$ be an endomorphism of R. An additive mapping H : R $\small{{\rightarrow}}$ R is called a left $\small{{\alpha}}$-multiplier (centralizer) if H(xy) = H(x)$\small{{\alpha}}$(y) holds for all x,y $\small{\in}$ R. In this paper, we shall investigate the commutativity of prime and semiprime rings admitting left $\small{{\alpha}}$-multiplier satisfying any one of the properties: (i) H([x,y])-[x,y] = 0, (ii) H([x,y])+[x,y] = 0, (iii) $\small{H(x{\circ}y)-x{\circ}y=0}$, (iv) $\small{H(x{\circ}y)+x{\circ}y=0}$, (v) H(xy) = xy, (vi) H(xy) = yx, (vii) $\small{H(x^2)=x^2}$, (viii) $\small{H(x^2)=-x^2}$ for all x, y in some appropriate subset of R.
Keywords
ideal;(semi)prime ring;generalized derivation;left multiplier (centralizer);left $\small{{\alpha}}$-multiplier;Jordan left $\small{{\alpha}}$-multiplier;
Language
English
Cited by
1.
Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings, International Journal of Mathematics and Mathematical Sciences, 2014, 2014, 1
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