Chapter

Commutative Algebra

pp 119-132

Date:

On a New Class of Integral Domains with the Portable Property

  • David E. DobbsAffiliated withDepartment of Mathematics, University of Tennessee
  • , Gabriel PicavetAffiliated withLaboratoire de Mathématiques, Université Blaise Pascal, UMR6620 CNRS, Les Cézeaux Email author 
  • , Martine Picavet-L’HermitteAffiliated withLaboratoire de Mathématiques, Université Blaise Pascal, UMR6620 CNRS, Les Cézeaux

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Abstract

A (commutative integral) domain R is said to be a pseudo-almost divided domain if for all P ∈ Spec(R) and uPR P , there exists a positive integer n such that u n P. Such domains are related to several known kinds of domains, such as divided domains and straight domains. It is shown that “locally pseudo-almost divided” is a portable property of domains. Hence, if T is a domain with a maximal ideal Q and D is a subring of TQ, then the pullback \(R:= T \times _{T/Q}D\) is locally pseudo-almost divided if and only if both T and D are locally pseudo-almost divided. A similar pullback transfer result is given for the “straight domain” property (which is not known to be portable) by imposing additional restrictions on the data T, Q, D.

Keywords

Integral domain Pullback Portable property Straight domain Pseudo-almost divided domain PAVD APVD Almost Prüfer domain Divided domain Root closed

Subject Classifications

[2010] Primary 13G05 Secondary 13A15 13F05 13B21