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Pseudo-valuation domains. (English) Zbl 0945.13012

From the paper: Let \(R\) be an integral domain with quotient field \(K\). A prime ideal \(P\) of \(R\) is said to be strongly prime if \(x,y \in K\) and \(xy\in P\) imply that \(x\in P\) or \(y\in P\). A domain \(R\) is called a pseudo-valuation domain (PVD) if each prime ideal of \(R\) is a strongly prime ideal. This concept of strongly prime ideals was introduced by J. R. Hedstrom and E. G. Houston [Pac. J. Math. 75, 137-147 (1978; Zbl 0368.13002)] in their study of pseudo valuation domains. In this short paper is generalize theorem 1.4 of the cited paper, that a quasi local domain is a PVD if and only if its maximal ideal is strongly prime.

MSC:

13F30 Valuation rings
13A15 Ideals and multiplicative ideal theory in commutative rings
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