Rings with a certain condition on subsemigroups (Q2367886)

A semigroup is a ring-semigroup iff it is the multiplicative semigroup of some ring. A subsemigroup \(S\) of the multiplicative semigroup of a ring \(R\) is called an \(a\)-subsemigroup iff the following conditions are satisfied: (i) \(S\) contains the zero of \(R\), (ii) \(a \in S\) implies \(-a \in S\). A ring \(R\) is an \(a\)-ring iff every \(a\)-subsemigroup of \(R\) is a ring-semigroup (note that an \(a\)-subsemigroup of a ring \(R\) need not to be a subring of \(R\)). Using some results of Satyanarayana and Cherubini and Varisco the author gives the full classification of \(a\)-rings.