\(E\)-solid rings, strongly IC rings and the Jacobson radical (Q6580186)

What properties of a semigroup \(S\) depend on the fact that \(S\) is the multiplicative semigroup of a ring \(R\), and conversely, what properties of \(R\) depend on those of \(S\)?\N\NIn this paper, the author investigates several such problems. For example, \(R\) is said to be \(E\)-solid if for any three idempotents \(e,\,f,\, g\), if \(eR=gR\) and \(Rg=Rf\), then there exists an idempotent \(h\) such that \(Re=Rh\) and \(hR=fR\), together with the dual property. The author shows that \(S\) is \(E\)-solid if and only if idempotents are central modulo the Jacobson radical of \(R\).\N\NThe paper contains two main theorems: the first is a list of ten properties of \(S\) which are shown to be equivalent if \(R\) is a unital ring, nine of which are equivalent if \(R\) lacks an identity element. The second is a proof of the equivalence of several ring theoretic properties of a not necessarily unital ring \(R\).