On rings with inverse adjoint semigroups
Let \(R\) be an associative ring. For all \(x,y\in R\) put \(x\circ y=x+y-xy\). Then \((R,\circ)\) is a semigroup, called the adjoint semigroup of \(R\). The operation \(\circ\) is well known in ring theory, e.g., \(R\) is Jacobson radical if and only if \((R,\circ)\) is a group. The author is interested in the case when \((R,\circ)\) is an inverse semigroup. More exactly, he considers the following problem. Let \(R\) be an associative ring, and let \(S\) be a semigroup; when has the semigroup ring \(RS\) an inverse adjoint semigroup? This problem is solved in two cases: for \(S\) commutative and for \(S\) a band (Theorems 1, 2). In both cases the solution is given in terms of the ideal of \(R\) generated by all idempotents of \(R\). The last result of this paper deals with band-graded rings. Suppose \(B\) is a band, and let \(R\) be a \(B\)-graded ring: \(R=\oplus R_b\) (\(b\in B\)) where every \(R_b\) is a subring of \(R\) with identity \(I_b\), \(R_a R_b\subseteq R_{ab}\), and \(I_aI_b=I_{ab}\) for all \(a,b\in B\) (so-called special \(B\)-graded ring). A ring \(L\) is strongly regular if and only if, for each \(x\in L\), there exists \(y\in L\) such that \(x=x^2y\). Suppose \(R=\oplus R_b\) (\(b\in B\)) is a special \(B\)-graded ring. Theorem 3 states that the following conditions are equivalent: (1) \((R,\circ)\) is inverse, (2) \((R,\circ)\) is a union of groups, (3) \(R\) is strongly regular, (4) all the components \(R_b\) are strongly regular and \(B\) is a semilattice.
scientific article; zbMATH DE number 1406415