Adjoint Clifford rings (Q1848171)

Let \(R\) be an associative ring and let \((R,\circ)\) be the adjoint semigroup of \(R\), where \(a\circ b=a+b-ab\) for \(a,b\in R\). This paper continues the authors' investigations into the relationship between a ring and its adjoint semigroup [\textit{H. Heatherly} and \textit{R. P. Tucci}, Acta Math. Hung. 90, No. 3, 231-242 (2001; Zbl 0973.20059)]. Here they study the case where \(R\) is adjoint Clifford, i.e. \((R,\circ)\) is a union of groups. If \((R,\circ)\) is a disjoint union of \(n\) (finite or infinite) groups, then \(R\) is said to be adjoint \(n\)-Clifford. Suppose that \(n<\infty\) or \(R\) satisfies a chain condition on the ideals. Then the main results of the paper assert that \(R\) is adjoint \(n\)-Clifford if and only if \[ R=D_1\oplus\cdots\oplus D_k\oplus J(R), \] where each \(D_i\) is a division ring, \(n=2^k\) and \(J(R)\) is the Jacobson radical of \(R\). Examples of adjoint Clifford rings are given which illustrate and delimit the theory involved.