Prüfer \(v\)-multiplication domains and related domains of the form \(D+D_S[\varGamma ^{*}]\)
Let \(D\) be an integral domain, \(S\) be a saturated multiplicative subset of \(D\) with \(D\subsetneq D_S\) and \(\Gamma\) be a nonzero torsion-free (additive) grading monoid with \(\Gamma\cap-\Gamma=\{0\}\). Let \(D_S[\Gamma]\) be the semigroup ring of \(\Gamma\) over \(D_S\), \(\Gamma^{*}=\Gamma-\{0\}\), and \(D^{(S,\Gamma)}=D+D_S[\Gamma^{*}]=\{f\in D_S[\Gamma]|f(0)\in D\}\). When \(\Gamma=\mathbb{N}_0\), the additive monoid of non-negative integers, then \(D^{(S,\mathbb{N}_0)}=D+XD_S[X]\), where \(X\) is an indeterminate, has received a considerable interest by several authors. In particular it is known that when \(D^{(S,\mathbb{N}_0)}\) is a P\(v\)MD (resp., GCD-domain, GGCD-domain). In the paper under review the authors determine when \(D^{(S,\Gamma)}\) is a P\(v\)MD (resp., GCD-domain, GGCD-domain) in this more general setting. In fact they prove that \(D^{(S,\Gamma)}\) is a P\(v\)MD (resp., GCD-domain, GGCD-domain) if and only if \(D\) is a P\(v\)MD (resp., GCD-domain, GGCD-domain), \(\Gamma\) is a valuation semigroup and \(S\) is a \(t\)-splitting (resp., splitting, \(d\)-splitting) set of \(D\).
torsion-free grading monoid \(\Gamma\) with \(\Gamma\cap-\Gamma=\{0\}\)
scientific article; zbMATH DE number 5718201
scientific article