Decompositions of multiplicative semigroups of m-domain rings and reduced Rickart rings

The main result of this article is that the multiplicative semigroup of an m-domain ring is a strong semilattice of certain subsemigroups, each of which turns out to be a \ monoid, and that this presentation of the semigroup as a strong semilattice of \ semigroups is essentially unique. As a consequence, it is shown that, given an m-domain ring R,+,· with the unary operation mapping every element to its minimal idempotent duplicator (in the sense of N.V.~Subrahmanyam), the algebra R,·, is a strong semilattice of \ s (in the sense of T.~Stokes), also essentially unique. Implications for reduced Rickart rings, which can be seen as a subclass of m-domain rings, are also described.