NOTE ON PSEUDO-VALUATION DOMAINS WHICH ARE NOT VALUATION DOMAINS

Year 2014, Volume: 16 Issue: 16, 53 - 65, 01.12.2014

Tariq Shah Waheed Ahmad Khan

https://doi.org/10.24330/ieja.266226

Abstract

In this article, we discuss the n-root closedness, root closedness,
seminormality, S-root closedness, S-closedness, F-closedess of PVDs. A valuation
domain, being integrally closed, is obviously root closed. So our interest
of study is for a class of non-valuation PVDs. Let R ⊂ B be a domain extension
such that R is a PVD and the common ideal P of R and B is a prime
ideal in R. If R is n-root closed (respectively root closed, seminormal, S-root
closed, S-closed, F-closed) in B, then R/P is PVD, which is n-root closed (respectively
root closed, seminormal, S-root closed, S-closed, F-closed) in B/P.
Further we study the relationship of atomic PVDs to atomic PVDs, SHFDs,
LHFDs and BVDs. We also discuss a relative ascent and descent in general
and particularly for the antimatter property of PVDs.

Keywords

PVD, atomic domain, F + M construction, root-closed, factor ring, condition ∗, antimatter domain

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• Department of Mathematics Quaid-I-Azam University Islamabad, Pakistan e-mail: [e-mail] Waheed Ahmad Khan Department of Mathematics and Statistics Caledonian College of Engineering P O Box 2322, Seeb 111, Sultanate of Oman e-mail: [e-mail]