Dedekind-Mertens lemma and content formulas in power series rings
Let \(R\) be a commutative ring with identity. For a polynomial \(f\in R[X]\), the ideal of \(R\) generated by the coefficients of \(f\) is denoted by \(A_f\) and called the content of \(f\). The best form of the Dedekind-Mertens lemma states that if \(f,g\in R[X]\), then \(A_f^{k+1}A_g=A_f^kA_{fg}\) where \(k+1\) is the locally minimal number of generators of \(A_g\). In particular, \(A_{fg}=A_fA_g\) for all \(f,g\in R[X]\) if and only if \(R\) is a Prüfer domain and \((A_{fg})_v=(A_fA_g)_v\) for all nonzero \(f,g\in R[X]\) if and only if \(R\) is integrally closed, where \(v\) is the \(v-\)operation on \(R\). For \(f\in RX\) a power series, we define the content of \(f\) as the ideal of \(R\) generated by the coefficients of \(f\). in the present paper, the authors show that if \(R\) is a Prüfer domain then \((A_fA_g)^2=(A_fA_g)A_{fg}\) for all \(f,g\in RX\). In the particular case where \(A_g\) is locally finitely generated, we have \(A^2_fA_g=A_fA_{fg}\) for all \(f\in RX\). As a consequence, an integral domain \(R\) is completely integrally closed if and only if \((A_fA_g)_v=(A _{fg})_v\) for all nonzero \(f,g\in RX\). Other related questions are answered.
scientific article; zbMATH DE number 6859161