Dedekind-Mertens lemma and content formulas in power series rings

Abstract

Let R[X] be the power series ring over a commutative ring R with identity. For f is an element of R[X], let A(f) denote the content ideal of f, i.e., the ideal of R generated by the coefficients of f. We show that if R is a Priifer domain and if g is an element of R[X] such that A(g) is locally finitely generated (or equivalently locally principal), then a Dedekind-Mertens type formula holds for g, namely A(f)(2)A(g) = A(f)A(fg) for all f is an element of R[X]. More generally for a Prefer domain R, we prove the content formula (A(f)A(g))(2) = (A(f)A(g))A(fg) for all f, g is an element of R[X]. As a consequence it is shown that an integral domain R is completely integrally closed if and only if (A(f)A(g))(v) = (A(fg))(v) for all nonzero f,g is an element of R[X], which is a beautiful result corresponding to the well-known fact that an integral domain R is integrally closed if and only if (A(f)A(g))(v) = (A(fg))(v) for all nonzero f,g is an element of R[X], where R[X] is the polynomial ring over R. For a ring R and g is an element of R[X], if A(g) is not locally finitely generated, then there may be no positive integer k such that A(f)(k+1)A(g) = A(f)(k)A(fg) for all f is an element of R[X]. Assuming that the locally minimal number of generators of A(g) is k +1, Epstein and Shapiro posed a question about the validation of the formula A(f)(k+1)A(g) = A(f)(k)A(fg) for all f is an element of R[X]. We give a negative answer to this question and show that the finiteness of the locally minimal number of special generators of A(g) is in fact a more suitable assumption. More precisely we prove that if the locally minimal number of special generators of A(g) is k + 1, then A(f)(k+1)A(g) = A(f)(k)A(fg) for all f is an element of R[X]. As a consequence we show that if A(g) is finitely generated (in particular if g is an element of R[X]), then there exists a nonnegative integer k such that A(f)(k+1)A(g)= A(f)(k)A(fg) for all f is an element of R[X]. (C) 2017 Elsevier B.V. All rights reserved.