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

Year 2024, Early Access, 1 - 24

Insa Cremer

https://doi.org/10.24330/ieja.1554197

Abstract

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.

Keywords

Reduced Rickart ring, m-domain ring, strong semilattice

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