TY - MGZN
AU - Giral, J.M.
AU - O'Carroll, L.
AU - Planas-Vilanova, F. A.
AU - Plans, B.
T2 - Proceedings of the Edinburgh Mathematical Society
Y1 - 2019
VL - 62
IS - 1
SP - 25
EP - 46
DO - 10.1017/S0013091518000275
UR - https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/on-the-integral-degree-of-integral-ring-extensions/4669AAC5255996C714C9CB482BE6FDE1
AB - 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.
TI - On the integral degree of integral ring extensions
ER -