Some results on pseudo-valuation domains (Q762555)

Let D be a domain, with quotient field K. An ideal P of D is strongly prime whenever: if x, \(y\in K\) and xy\(\in P\) then \(x\in P\) or \(y\in P\). If every prime ideal of D is strongly prime, D is called a pseudo- valuation domain (PVD). Every valuation domain is a PVD, every PVD D has only one maximal ideal M and \((D:M)_ K=\{x\in K| xM\subseteq D\}\) is a valuation domain V, \(V\supseteq D\), with maximal ideal M. This definition and results were given by Hedstrom and Houston. In the present paper, the author indicates more properties of PVD's. For example: (1) if \(P\subseteq I\) are ideals of a PVD D and P is prime then P is also a prime ideal of \((I:I)_ K\); (2) P is the unique maximal ideal of \((P:P)_ K\); (3) if \(P\neq M\) then \((P:P)_ K=D_ P\) and \(P=(D:D_ P)_ D=\{d\in D| dD_ P\subseteq D\};\) (4) if P is properly contained in the ideal \(I\neq D\) then \(P\subseteq \cap^{\infty}_{n=1}I^ n;\) (5) the mapping \(D'\to (P:D')_ D=\{d\in D| dD'\subseteq P\}\) establishes a bijection between the set of all overrings of D properly containing \((P:P)_ K\) and the set of all prime ideals P' of D, properly contained in P. An example shows that (5) is not true if D is not assumed to be a PVD.