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 \rcancellative\ monoid, and that this presentation of the semigroup as a strong semilattice of \rcancellative\ semigroups is essentially unique. As a consequence, it is shown that, given an m-domain ring $ \ang{R,+,\cdot} $ with the unary operation $ \dop{} $ mapping every element to its minimal idempotent duplicator (in the sense of N.V.~Subrahmanyam), the algebra $ \ang{R,\cdot,\dop{}} $ is a strong semilattice of \rcancellative\ \dsemigroup 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.