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On the integral degree of integral ring extensions
- Source :
- Recercat. Dipósit de la Recerca de Catalunya, instname, UPCommons. Portal del coneixement obert de la UPC, Universitat Politècnica de Catalunya (UPC)
- Publication Year :
- 2019
-
Abstract
- 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.
- Subjects :
- Noetherian
Pure mathematics
General Mathematics
Dedekind domain
Matemàtiques i estadística [Àrees temàtiques de la UPC]
Mathematics - Commutative Algebra
Commutative Algebra (math.AC)
Integral equation
Nagata ring
13B21, 13B22, 13G05, 12F05
Àlgebra commutativa
Integrally closed
Field extension
FOS: Mathematics
Algebraic number
Invariant (mathematics)
Commutative algebra
Mathematics
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- Recercat. Dipósit de la Recerca de Catalunya, instname, UPCommons. Portal del coneixement obert de la UPC, Universitat Politècnica de Catalunya (UPC)
- Accession number :
- edsair.doi.dedup.....55301f5ad5539ac4e70d260c098dc37d