Every divisor class of Krull monoid domains contains a prime ideal
Let \(\Lambda\) be an index set of given cardinality, and let \(G_0 = \sum_{i\in \Lambda}\mathbb{Z}_i\), where each \(\mathbb{Z}_i\) is the additive group of integers. Let \(F\) be a field and \(X_i, Y_i , T_i , U_i\) be indeterminates over \(F\) with \(X_iU_i = Y_iT_i\), and let \(A = F[\{X_i, Y_i, T_i, U_i\}_{i\in \Lambda}]\). \textit{L. Claborn} [Pac. J. Math. 18, 219--222 (1966; Zbl 0166.30602)] showed that \(A\) is a Krull domain with \(Cl(A) = G_0\). He also showed that each divisor class of \(A[X]\), the polynomial ring over \(A\) over \(X\), contains a prime ideal, and that every abelian group is the divisor class group of a Krull domain. The author in the paper under review shows that every divisor class of Krull monoid domains contains a prime ideal. Let \(D\) be an integral domain with quotient field \(K\), \(\Gamma\) be a nontrivial torsion-free grading monoid with quotient group \(G\) and \(D[\Gamma]\) be the monoid domain of \(\Gamma\) over \(D\). Let \(F(D)\) be the set of nonzero fractional ideals of \(D\). For each \(I\in F(D)\), let \(I^{-1} = \{x\in K | xI\subseteq D\}\), \(I_v = {(I^{-1})}^{-1}\) and \(I_t = \bigcup \{J_v | J\subseteq I \text{ and }J\) is a nonzero finitely generated ideal
scientific article; zbMATH DE number 5996424
scientific article