Abstract
All rings considered are commutative with identity. Let R be a complemented ring with integral closure R0 (in its total quotient ring K). Then R ⊆ S satisfies INC for each overring S of R (inside K) if and only if R0 is a Prüfer ring. If R0 is a Prüfer ring and T is a complemented ring that contains R as a subring such that each regular element of T is a root of a polynomial in R[X] with a regular coefficient and T is torsion-free over R, then R ⊆ T satisfies INC. As a consequence, a new generalization for rings with nontrivial zero-divisors is found of Pr¨ufer’s result on the integral closure of a Prüfer domain in a field extension of the quotient field.
Keywords
complemented ring INC Pr¨ufer ring ring extension overring integral algebraic von Neumann regular ring integral domain torsion-free total quotient ring flat
References
- D. F. Anderson and A. Badawi, Divisibility conditions in commutative rings with zerodivisors, Comm. Algebra, 30 (2002), 4031–4047.
- D. E. Dobbs, On INC-extensions and polynomials with unit content, Canad. Math. Bull., 23 (1980), 37–42.
- D. E. Dobbs, Pr¨ufer’s ascent result via INC, Comm. Algebra, 23 (1995), –119.
- S. Endo, On semi-hereditary rings, J. Math. Soc. Japan, 13 (1961), 5413–
- R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972.
- M. Griffin, Pr¨ufer rings with zero divisors, J. Reine Angew. Math., 239/240 (1969), 55–67.
- J. A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York, 1988.
- I. Kaplansky, Commutative Rings, rev. ed., Univ. Chicago Press, Chicago, I. J. Papick, Topologically defined classes of going-down domains, Trans. Amer. Math. Soc., 219 (1976), 1–37.
- G. Picavet and M. Picavet-L’Hermitte, Seminormal or t-closed schemes and Rees rings, Algebr. Represent. Theory, 1 (1998), 255–309.
- H. Pr¨ufer, Untersuchungen ¨uber Teilbarkeitseigenschaften in K¨orpern, J. Reine Angew. Math., 168 (1932), 1–36.
- F. Richman, Generalized quotient rings, Proc. Amer. Math. Soc.,16 (1965), –799.
- H. Uda, Incomparability in ring extensions, Hiroshima Math. J., 9 (1979), –463. David E. Dobbs
- Department of Mathematics University of Tennessee Knoxville, TN 37996-0612, U.S.A. e-mail: dobbs@math.utk.edu Jay Shapiro
- Department of Mathematical Sciences George Mason University Fairfax, VA 22030-4444, U.S.A. e-mail: jshapiro@gmu.edu
Abstract
References
- D. F. Anderson and A. Badawi, Divisibility conditions in commutative rings with zerodivisors, Comm. Algebra, 30 (2002), 4031–4047.
- D. E. Dobbs, On INC-extensions and polynomials with unit content, Canad. Math. Bull., 23 (1980), 37–42.
- D. E. Dobbs, Pr¨ufer’s ascent result via INC, Comm. Algebra, 23 (1995), –119.
- S. Endo, On semi-hereditary rings, J. Math. Soc. Japan, 13 (1961), 5413–
- R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972.
- M. Griffin, Pr¨ufer rings with zero divisors, J. Reine Angew. Math., 239/240 (1969), 55–67.
- J. A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York, 1988.
- I. Kaplansky, Commutative Rings, rev. ed., Univ. Chicago Press, Chicago, I. J. Papick, Topologically defined classes of going-down domains, Trans. Amer. Math. Soc., 219 (1976), 1–37.
- G. Picavet and M. Picavet-L’Hermitte, Seminormal or t-closed schemes and Rees rings, Algebr. Represent. Theory, 1 (1998), 255–309.
- H. Pr¨ufer, Untersuchungen ¨uber Teilbarkeitseigenschaften in K¨orpern, J. Reine Angew. Math., 168 (1932), 1–36.
- F. Richman, Generalized quotient rings, Proc. Amer. Math. Soc.,16 (1965), –799.
- H. Uda, Incomparability in ring extensions, Hiroshima Math. J., 9 (1979), –463. David E. Dobbs
- Department of Mathematics University of Tennessee Knoxville, TN 37996-0612, U.S.A. e-mail: dobbs@math.utk.edu Jay Shapiro
- Department of Mathematical Sciences George Mason University Fairfax, VA 22030-4444, U.S.A. e-mail: jshapiro@gmu.edu
Details
| Other ID | JA68KH83YB |
|---|---|
| Journal Section | Articles |
| Authors | |
| Publication Date | June 1, 2010 |
| Published in Issue | Year 2010 Volume: 7 Issue: 7 |