Notes on M-semigroups
Let \(S=(S,+)\) be a torsion-free cancellative commutative semigroup with zero, and let G be the quotient group of S, \(G\neq S\). If \(A\subseteq S\) then \(A^{-1}=\{x\in G:\) \(x+A\subseteq G\}\) and \((A^{-1})^{-1}=A^ v\). If A is an ideal of S with \(A=A^ v\) then A is a v-ideal of S. S is a Mori-semigroup iff S satisfies ACC on v-ideals. A Mori-semigroup S is called an M-semigroup iff any ideal of S generated by two elements is a v-ideal. If each ideal of S is a v-ideal then S is said to be reflexive. By dim S the Krull dimension of S is denoted, i.e. the maximal integer n such that there exists a chain \(P_ 1\supset...\supset P_ n\) of prime ideals of S. If dim S\(=1\) then there exists a unique maximal ideal of S denoted by M. The main theorem of the paper states that, for any Mori- semigroup S, the following conditions are equivalent: (1) dim S\(=1\) and \(M^{-1}\) is generated by two elements, (2) S is reflexive, (3) S is an M-semigroup. In fact, in the paper the implication (1)\(\Rightarrow (2)\) is proved only, since the other implications (3)\(\Leftrightarrow (2)\Rightarrow (1)\) were established in an earlier paper [\textit{K. Sato} and \textit{R. Matsuda}, On M-semigroups, Mem. Tohoku Inst. Technol., Ser. I 8, 1-7 (1988)].
torsion-free cancellative commutative semigroup
scientific article; zbMATH DE number 4190040