On additive maps of MA-semirings with involution

AbstractLet R be a ring. A map f: R → R is additive if f(a + b) = f(a) + f(b) for all elements a and b of R. Here, a map f: R → R is called unit-additive if f(u + v) = f(u) + f(v) for all units u and v of R. Motivated by a recent result of Xu, Pei and Yi showing that, for any field F, every unit-additive map of (F) is additive for all n ≥ z, this paper is about the question of when every unit-additivemap of a ring is additive. It is proved that every unit-additivemap of a semilocal ring R is additive if and only if either R has no homomorphic image isomorphic to or R/J(R) ≅ with 2 = 0 in R. Consequently, for any semilocal ring R, every unit-additive map of (R) is additive for all n ≥ 2. These results are further extended to rings R such that R/J(R) is a direct product of exchange rings with primitive factors Artinian. A unit-additive map f of a ring R is called unithomomorphic if f(uv) = f(u)f(v) for all units u, v of R. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

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COMPOUND-COMMUTING ADDITIVE MAPS ON MATRIX SPACES

Journal of the Korean Mathematical Society ◽

10.4134/jkms.2011.48.1.083 ◽

2011 ◽

Vol 48 (1) ◽

pp. 83-104 ◽

Cited By ~ 1

Author(s):

Wai Leong Chooi

Keyword(s):

Matrix Spaces ◽

Additive Maps

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Additive Maps Preserving Local Spectrum

Integral Equations and Operator Theory ◽

10.1007/s00020-005-1392-2 ◽

2005 ◽

Vol 55 (3) ◽

pp. 377-385 ◽

Cited By ~ 22

Author(s):

Abdellatif Bourhim ◽

Thomas Ransford

Keyword(s):

Local Spectrum ◽

Additive Maps

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Additive maps having a generalized derivation expansion

Journal of Algebra and Its Applications ◽

10.1142/s0219498815500486 ◽

2015 ◽

Vol 14 (04) ◽

pp. 1550048 ◽

Cited By ~ 3

Author(s):

Tsiu-Kwen Lee

Keyword(s):

Central Simple Algebra ◽

Simple Algebra ◽

Generalized Derivation ◽

Elementary Operator ◽

Additive Map ◽

Finite Dimensional ◽

Left And Right ◽

Central Simple ◽

Skolem Theorem ◽

Additive Maps

Let R be a prime ring with extended centroid C. We prove that an additive map from R into RC + C can be characterized in terms of left and right b-generalized derivations if it has a generalized derivation expansion. As a consequence, a generalization of the Noether–Skolem theorem is proved among other things: A linear map from a finite-dimensional central simple algebra into itself is an elementary operator if it has a generalized derivation expansion.

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