Dedekind-Mertens lemma and content formulas in power series rings

For a ring R and g∈R〚X〛, if Ag is not locally finitely generated, then there may be no positive integer k such that Afk+1Ag=AfkAfg for all f∈R〚X〛. Assuming that the locally minimal number of generators of Ag is k+1, Epstein and Shapiro posed a question about the validation of the formula Afk+1Ag=AfkAfg for all f∈R〚X〛. We give a negative answer to this question and show that the finiteness of the locally minimal number of special generators of Ag is in fact a more suitable assumption. More precisely we prove that if the locally minimal number of special generators of Ag is k+1, then Afk+1Ag=AfkAfg for all f∈R〚X〛. As a consequence we show that if Ag is finitely generated (in particular if g∈R[X]), then there exists a nonnegative integer k such that Afk+1Ag=AfkAfg for all f∈R〚X〛.