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Generalized adjoint semigroups of a ring. (English) Zbl 1108.16017

A binary operation \(\diamondsuit \) on a ring \(R\) is called a generalized adjoint multiplication on \(R\) if it satisfies the following three conditions: (i) the associative law, (ii) two generalized distributive laws: \(x\diamondsuit(y+z)=x\diamondsuit y+x\diamondsuit z-x\diamondsuit 0\) and \((y+z)\diamondsuit x=y\diamondsuit x+z\diamondsuit x-0\diamondsuit x\), and (iii) the compatibility: \(xy=x\diamondsuit y-x\diamondsuit 0-0\diamondsuit y+0\diamondsuit 0\). The semigroup \((R,\diamondsuit)\) is called a generalized adjoint semigroup of \(R\) or, in short, GA-semigroup and is denoted by \(R^\diamondsuit\). This is a generalization of the multiplicative semigroup \(R^\bullet\) and the adjoint semigroup \(R^\circ\) of a ring \(R\). A bitranslation is a pair \((\lambda,\rho)\in\text{End}(R_R)\times\text{End}({_RR})\) such that \(x\lambda(y)=\rho(x)y\) for any \(x,y\in R\).
In this paper the authors describe generalized adjoint semigroups of a ring \(R\). In Section 2, they show how to construct generalized adjoint multiplications on a ring \(R\) by means of bitranslations of \(R\), characterize a GA-semigroup with identity or zero and describe GA-semigroups of a ring with 1. In Section 3, the authors prove that GA-semigroups of a \(\pi\)-regular ring are \(\pi\)-regular. In Section 4, they show that a GA-semigroup containing idempotents can be represented as a GA-semigroup of the ring of a Morita context. They also give a necessary and sufficient condition for a GA-semigroup to contain idempotents and prove that in any ring, idempotents can be lifted modulo a \(\pi\)-regular ideal. This generalizes a classical result in ring theory which states that idempotents modulo a nil ideal can be lifted. Finally the authors show that GA-semigroups of rings with DCC on principal right ideals contain idempotents.

MSC:

16N20 Jacobson radical, quasimultiplication
20M25 Semigroup rings, multiplicative semigroups of rings
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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