Journal article
The primitive element theorem for commutative algebras
Houston Journal of Mathematics, Vol.25(4), pp.603-623
1999
Abstract
Let R ⊆ T be an extension of commutative rings (with the same 1). We say that R ⊆ T has FIP if the set of R-subalgebras of T is finite. If R ⊆ T has FIP, then T must be algebraic over R; if, in addition, R is a field, then T is a finite-dimensional R-vector space. If R ⊆ T has FIP and T is an integral domain, then either R and T are fields or T is an overring of R. If R is a perfect field, then the main result identifies four exhaustive cases which serve to characterize the condition that R ⊆ T has FIP. Considering extensions R ⊆ T having FIP with T the quotient field of R amounts to studying integral domains R with only finitely many overrings. Such integral domains R are characterized as the semi-quasilocal i-domains of finite Krull dimension having only finitely many integral overrings. This property is interpreted further in case R is either integrally closed or a pseudo-valuation domain. Examples are given to illustrate the sharpness of the results.
Details
- Title: Subtitle
- The primitive element theorem for commutative algebras
- Creators
- D.D. AndersonD.E. DobbsB. Mullins
- Resource Type
- Journal article
- Publication Details
- Houston Journal of Mathematics, Vol.25(4), pp.603-623
- ISSN
- 0362-1588
- Language
- English
- Date published
- 1999
- Academic Unit
- Mathematics
- Record Identifier
- 9984230628502771
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