Title: The Dedekind-Mertens Theorem for power series rings and a content function for arbitrary algebras over a ring
Speaker: Jay Shapiro, George Mason University
Abstract: In 1892 two mathematicians, R. Dedekind and F. Mertens, independently proved a generalization of Gauss' Lemma that works for any commutative ring R. Specifically, they proved that if f,g \in R[x], then there exists an integer k such that [c(f)^k][c(g)]=[c(f)^{k-1}][c(fg)], where c(h) denotes the content of the polynomial h. We generalize this formula to power series over a commuta- tive Noetherian ring R, where the notion of content extends in the obvious fashion (this also corrects an error in the literature). In the second half of the talk we examine the notion of a content function \Omega that J. Ohm and D. Rush generalized to an arbitrary algebra over a ring (though it can behave quite badly). We examine certain properties an algebra may have with respect to this function - content algebra, weak content algebra, and semicontent algebra (our own definition). Finally, we consider conditions when the Dedekind-Mertens formula holds for \Omega for power series over a valuation ring.
