A visit to valuation and pseudo-valuation domains. (English) Zbl 0885.13015

Badawi, Ayman

Anderson, David F. (ed.) et al., Zero-dimensional commutative rings. Proceedings of the 1994 John H. Barrett memorial lectures and conference on commutative ring theory, Knoxville, TN, USA, April 7–9, 1994. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 171, 155-161 (1995).

Throughout this paper, \(R\) denotes a commutative domain with \(1\neq 0\) and \(K\) denotes the quotient field of \(R\). Recall [see J. R. Hedstrom and E. G. Houston, Pac. J. Math. 75, 137-147 (1978; Zbl 0368.13002)] that a prime ideal \(P\) of \(R\) is called strongly prime if \(x,y\in K\) and \(xy\in P\) imply that \(x\in P\) or \(y\in P\). If every prime ideal of \(R\) is strongly prime, then \(R\) is called a pseudo-valuation domain (abbreviated PVD). In this paper, we give alternative proofs of some well-known results in the paper cited above and a paper by D. F. Anderson [Houston J. Math. 9, 325-332 (1983; Zbl 0526.13015)]. Let \(P\) be a nonzero strongly prime ideal of \(R\). If \(P\) contains a prime element of \(R\), then we show that \(P\) is a principal maximal ideal of \(R\) and \(R\) is a valuation domain. Furthermore, we give an alternative proof of the fact that \(P^{-1}= (P:P)= \{x\in K:x P\subset P\}\) is a ring and we give a more general version of this fact.
For the entire collection see [Zbl 0872.00033].

MSC:

13F30	Valuation rings
13G05	Integral domains

Keywords:

strongly prime ideal; pseudo-valuation domain

Citations:

Zbl 0368.13002; Zbl 0376.13001; Zbl 0526.13015