Finitely generated multiplicative subsemigroups of rings (Q1375879)

Let \(R\) be a commutative ring (not necessarily with identity). In [Proc. Am. Math. Soc. 10, 908-909 (1959; Zbl 0092.27301)], \textit{J. R. Isbell} showed that if the multiplicative semigroup \((R,\cdot)\) is finitely generated, then \(R\) is finite. Let \(Z(R)\), \(\text{reg}(R)\) and \(U(R)\) denote the sets of zero divisors, regular elements, and units of \(R\), respectively. The main result of this paper is that there is a finitely generated subsemigroup \((S,\cdot)\) of \((R,\cdot)\) with \(R-\text{reg}(R)=Z(R)\subset S\) (resp., \(R-U(R)\subset S\)) if and only if either (1) \(R\) is finite, or (2) \(R\) is an integral domain (resp., a field).