Content formulas for power series and Krull domains (Q501815)

In this article, the authors are interested in generalizing and extending much of the work on content formulas for power series by studying \(*\) operations for Prüfer \(\nu\)-multiplication domains (PVMDs), Krull domains and Dedekind domains. In many ways, the paper is motivated by and seeks to build on the work of \textit{D. D. Anderson} and \textit{B. G. Kang} [J. Algebra 181, No. 1, 82--94 (1996; Zbl 0857.13017)] as well as other related work where similar formulas were developed. Let \(R\) be an integral domain with quotient field \(K\) and let \(F(R)\) be the set of non-zero fractional ideals of \(R\). Let \(f \in K[[X]][[X^{-1}]]\), then the content \(c(f)\) is called the content of \(f\) and is defined to be the \(R\)-submodule of \(K\) generated by the coefficients of \(f\). A star operation, \(*:I \to I_*\) is a map from \(F(R)\) into \(F(R)\) such that for all \(0 \neq a \in K\) and all \(A,B \in F(R)\), \((1) (a)=(a)_*\), \((aA)_*=aA_*\), (2) \(A \subseteq A_*\) and \(A \subseteq B\) implies \(A_* \subseteq B_*\), and (3) \((A_*)_*=A_*\). The most well known examples of star operations include \(d\)-, \(\nu\)-, \(t\)-, and \(\omega\)- which are \(A_d=A\) (the identity operation), \(A_\nu=(A^{-1})^{-1}\), \(A_t= \cup B_\nu\) where \(B\) ranges over the finitely generated subideals of \(A\), and \(A_\omega=\{x \in K \mid Jx \subseteq A \) for some finitely generated ideal \(J\) with \(J^{-1}-R\}\), respectively. There are many results proved in the article; however, the main results include, but are not limited to the following. \(R\) is a Krull domain if and only if \(c(f/g)_\omega=(c(f)c(g)^{-1})_\omega\) for all non-zero \(f,g \in R[[X]]\) with \(c(f/g)\) a fractional ideal if and only if \(c(f/g)_t=(c(f)c(g)^{-1})_t\) for all non-zero \(f,g \in R[[X]]\) with \(c(f/g)\) a fractional ideal and \(R\) is a Dedekind domain if and only if for all non-zero \(f,g \in R[[X]]\) with \(c(f/g)\) a fractional ideal, \(c(f/g)=c(f)c(g)^{-1}\).