ON THE MAXIMAL CARDINALITY OF CHAINS OF INTERMEDIATE RINGS

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

References

1. D. D. Anderson, D. E. Dobbs and B. Mullins, The primitive element theorem for commutative algebras, Houston J. Math., 25 (1999), 603–623. Corrigen- dum, Houston J. Math., 28 (2002), 217–219.
2. A. Ayache and A. Jaballah, Residually algebraic pairs of rings, Math. Zeit., (1997), 49–65.
3. E. Curtin, Infinite rings whose subrings are nested, Proc. Roy. Irish Acad. Sect. A, 94 (1994), 59–66.
4. E. Curtin, Finite rings whose subrings are nested, Proc. Roy. Irish Acad. Sect. A, 94 (1994), 67–75.
5. D. E. Dobbs, Extensions of integral domains with infinite chains of interme- diate rings, Comm. Algebra, to appear. D. E. Dobbs, B. Mullins, G. Picavet and M. Picavet-L’Hermitte, On the FIP property for extensions of commutative rings, Comm. Algebra, 33 (2005), –3119.
6. D. E. Dobbs, B. Mullins and M. Picavet-L’Hermitte, The singly generated uni- tal rings with only finitely many unital subrings, Comm. Algebra, 36 (2008), –2653.
7. D. E. Dobbs, G. Picavet and M. Picavet-L’Hermitte, A characterization of the commutative unital rings with only finitely many unital subrings, J. Algebra Appl., to appear. D. E. Dobbs and J. Shapiro, A classification of the minimal ring extensions of an integral domain, J. Algebra, 305 (2006), 185–193.
8. M. S. Gilbert, Extensions of commutative rings with linearly ordered inter- mediate rings, Ph. D. dissertation, University of Tennessee, Knoxville, TN, R. Gilmer, Some finiteness conditions on the set of overrings of an integral domain, Proc. Amer. Math. Soc., 131 (2003), 2337–2346.

9. G. Picavet and M. Picavet-L’Hermitte, About minimal morphisms, Multi- plicative Ideal Theory in Commutative Algebra, Springer-Verlag, New York (2006), 369–386.
10. J. Sato, T. Sugatani and K. I. Yoshida, On minimal overrings of a Noetherian domain, Comm. Algebra, 20 (1992), 1735–1746. David E. Dobbs
11. Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, U.S.A. e-mail: [e-mail]
12. 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]