Pseudo-dedekind rings, semistar operations, and content formulas for power series

Abstract

We prove that an integral domain D is pseudo-Dedekind if and only if it satisfies a certain “Gauss’s lemma for power series”: c ( f g ) v = c ( f ) v c ( g ) v for all power series f and g over D in any (possibly infinite) number of indeterminates, where c ( f ) v denotes the v-closure of the content of f. We generalize this fact via semistar operations, coefficient rings with zero divisors, and fractional (semi)regular contents. Along the way, we prove several other new facts about (semistar) power series content formulas, generalizing as above on known results about Prüfer (v-multiplication) domains, (completely) integrally closed domains, and (generalized) GCD domains.