Bruce Olberding

Commutative algebra : noetherian and non-noetherian perspectives

- Introduction.- 1. Principal-like Ideals and Related Polynomial Content Conditions (D.D. Anderson).- 2. Zero-divisor Graphs in Commutative Rings (D.F. Anderson, M. Axtell, J. Stickles).- 3. Class Semigroups and T-class Semigroups of Domains (S. Bazzoni, S. Kabbaj).- 4. Forcing Algebras, Syzygy Bundles, and Tight Closure (H. Brenner).- 5. Beyond Totally Reflexive Modules and Back: A Survey on Gorenstein Dimensions (L.W. Christensen, H.B. Foxby, H. Holm).- 6. On V-domains: A Survey (M. Fontana, M. Zafrullah).- 7. Tensor Products of Algebras Over a Field (H. Haghighi, M. Tousi, S. Yassemi).- 8. Multiplicative Ideal Theory in the Context of Commutative Monoids (F. Halter-Koch).- 9. Projectively-Full Ideals and Compositions of Consistent Systems of Rank One Discrete Valuation Rings: A Survey (W. Heinzer, J. Ratliff, D. Rush).- 10. Direct-sum Behavior of Modules Over One-dimensional Rings (R. Karr, R. Wiegand).- 11. The Defect (F.V. Kuhlmann).- 12. The Use of Ultrafilters to Study the Structure of a Commutative Ring (K.A. Loper).- 13. Integrally Closed Overrings of Two-dimensional Noetherian Domains (B. Olberding).- 14. Almost Perfect Domains and Their Modules (L. Salce).- 15. Characteristic p Methods in Characteristic Zero via Ultraproducts (H. Schoutens).- 16. Rees Valuations (I. Swanson).- 17. Weak Normality and Seminormality (M. Vitulli).

Divisibility properties in ultrapowers of commutative rings

All rings considered are commutative with identity. We study the preservation of certain properties in the passage from a ring R to the ultrapower R relative to a free ultrafilter on the set of all positive integers. Our main result is that if R is a locally pseudo-valuation domain (LPVD) of finite character (for instance, a semi-quasilocal LPVD), then R is also an LPVD. In the same vein, it is shown that the classes of pseudo-valuation domains and pseudo-valuation rings are each stable under the passage from R to R . An example is given of a divided domain R such that the domain R is not divided. A divisibility condition is found which characterizes the divided (respectively, quasilocal) rings R such that R is a divided (respectively, treed) ring.