Commutative rings with finitely generated monoids of fractional ideals
Let \(R\) be a commutative ring with integral closure \({\bar R}\), total quotient ring T(R), and let \({\bar F}(R)\) be the multiplicative monoid consisting of all \(R\)-submodules of \(T(R)\). The purpose of the paper is to investigate the finite generation property for \({\bar F}(R)\) or one of the following submonoids of \({\bar F}(R)\): the set of principal fractional ideals \(P(R)\), the set of finitely generated fractional ideals \(F^*(R)\), and the set of fractional ideals \(F(R)\). Recall the following definitions: the group of divisibility \(G(R)\) of \(R\) is \(U(T(R))/U(R)\) and a finite dimensional semiquasilocal generalized Dedekind domain is a semiquasilocal domain \(D\) such that for each maximal ideal \(M\) of \(D\), \(D_M\) is a finite-rank discrete valuation domain. A special principal ideal ring (SPIR) is a principal ideal ring with a unique nonzero prime ideal, which is nilpotent. A special Bézout ring \(R\) is a Bézout ring (every finitely generated ideal of \(R\) is principal) with a unique minimal prime \(P\), \(T(R) = R_P\) is an SPIR, and \(G(R/P)\) is finitely generated. Then the following conditions are equivalent: (i) \(P(R)\) is finitely generated. (ii) \(G(R)\) is finitely generated and \(T(R)\) is a finite direct product of finite local rings, SPIRs, and fields. (iii) \(F^*(R)\) is finitely generated. (iv) \({\bar R}/[R : {\bar R}]\) is finite and \({\bar R}\) is a finite direct product of finite local rings, special Bézout rings, and Bézout domains with finitely generated groups of divisibility. Another result says that the following conditions are equivalent. (1) \(F(R)\) is finitely generated. (2) \({\bar F}(R)\) is finitely generated. (3) \({\bar R}/[R : {\bar R}]\) is finite and \({\bar R}\) is a finite direct product of finite local rings, special Bézout rings whose reduced rings are finite dimensional semiquasilocal generalized Dedekind domains, and finite dimensional semiquasilocal generalized Dedekind domains.
scientific article; zbMATH DE number 5376943