Jordan Homomorphisms of Rings

Abstract

The primary aim of this paper is to study mappings J of rings that are additive and that satisfy the conditions

$$ {\left( {{a^2}} \right)^J} = {\left( {{a^J}} \right)^2},\;{\left( {aba} \right)^J} = {a^J}{b^J}{a^J} $$

(1)

Such mappings will be called Jordan homomorphisms. If the additive groups admit the operator 1/2 in the sense that 2x = a has a unique solution (1/2)a for every a, then conditions (1) are equivalent to the simpler condition

$$ {\left( {ab} \right)^J} + {\left( {ba} \right)^J} = {a^J}{b^J} + {b^J}{a^J} $$

(2)

Mappings satisfying (2) were first considered by Ancochea [1], [2](¹). The modification to (1) is essentially due to Kaplansky [13]. Its purpose is to obviate the necessity of imposing any restriction on the additive groups of the rings under consideration.