On the integral degree of integral ring extensions

Let A ¿ B be an integral ring extension of integral domains with fields of fractions K and L, respectively. The integral degree of A ¿ B, denoted by dA(B), is defined as the supremum of the degrees of minimal integral equations of elements of B over A. It is an invariant that lies in between dK(L) and µA(B), the minimal number of generators of the A-module B. Our purpose is to study this invariant. We prove that it is sub-multiplicative and upper-semicontinuous in the following three cases: if A ¿ B is simple; if A ¿ B is projective and finite and K ¿ L is a simple algebraic field extension; or if A is integrally closed. Furthermore, d is upper-semicontinuous if A is noetherian of dimension 1 and with finite integral closure. In general, however, d is neither sub-multiplicative nor upper-semicontinuous.Peer ReviewedPostprint (author's final draft