Regular generalized adjoint semigroups of a ring. (Q2488610)

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 \(R\). The authors prove that a GA-semigroup with central idempotents is a product of a multiplicative semigroup and an adjoint semigroup of ideals. They determine GA-semigroups of a strongly regular ring. Moreover, they describe rings with a GA-semigroup having a property \(\mathbf P\) and its such GA-semigroups in terms of the ring of a Morita context, where \(\mathbf P\) denotes orthodox, right inverse, inverse, pseudoinverse, \(E\)-unitary, and completely simple, respectively. In particular, they observe that \(R^\bullet\) has the property \(\mathbf P\) implies \(R^\diamondsuit\) has the property \(\mathbf P\) implies \(R^\circ\) has the property \(\mathbf P\).