Weighted Jordan homomorphisms
Matej Bre\v{s}ar, Maria Luisa C. Godoy

TL;DR
This paper characterizes weighted Jordan homomorphisms in unital rings, especially matrix rings, by identifying conditions under which additive maps preserving certain zero products are such homomorphisms.
Contribution
It establishes new conditions under which additive maps are weighted Jordan homomorphisms, extending known results to broader classes of rings including matrix and prime rings.
Findings
Surjective maps preserving zero products are weighted Jordan homomorphisms under certain conditions.
Bijective maps on prime rings are weighted Jordan homomorphisms if related to a map satisfying a quadratic condition.
Results apply to matrix rings over rings with 1/2, generalizing previous characterizations.
Abstract
Let and be unital rings. An additive map is called a weighted Jordan homomorphism if is an invertible central element and for all . We provide assumptions, which are in particular fulfilled when with and any unital ring with , under which every surjective additive map with the property that whenever is a weighted Jordan homomorphism. Further, we show that if is a prime ring with char, then a bijective additive map is a weighted Jordan homomorphism provided that there exists an additive map such that for all .
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Taxonomy
TopicsAdvanced Topics in Algebra · Rings, Modules, and Algebras
