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 d _A (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 d _K (L) and μ _A (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. [ABSTRACT FROM AUTHOR]