ON THE MAXIMAL CARDINALITY OF CHAINS OF INTERMEDIATE RINGS

Year 2009, Volume: 5 Issue: 5, 121 - 134, 01.06.2009

David E. Dobbs Gabriel Picavet Martine Picavet-l’hermitt

Abstract

All rings considered are commutative with 1 and all subrings are unital. If R ⊆ T are rings such that T is a finitely generated R-module, R is not a total quotient ring and (R : T) = 0, then there exists a denumerable chain of R-subalgebras of T. The rings having only finite chains of subrings are shown to be the same as the recently classified rings having only finitely many subrings.

Keywords

Commutative ring, ring extension, subalgebra, chain, cardinality, determinant, FIP property, FSP property, integrality, greatest common divisor, pullback, singly generated ring, integral domain, total quotient ring

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• Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, U.S.A. e-mail: [e-mail]
• Gabriel Picavet and Martine Picavet-L’Hermitte Laboratoire de Math´ematiques Pures Universit´e Blaise Pascal Aubi`ere Cedex, France e-mails: [e-mail] [e-mail]