Note on pseudo-valuation domains which are not valuation domains (Q2879410)

A pseudo-valuation domain (PVD) is a quasi-local integral domain \(R\) whose maximal ideal \(M\) satisfies: \(\forall(x,y)\in K\times K\), \(xy\in M\Rightarrow x\in M\) or \(y\in M\) (we say that \(M\) is strongly prime); if this holds then every prime ideal of \(R\) is strongly prime. If \(R\) is a PVD, then the set \(V= \{x\in K\mid xM\subset M\}\) is a valuation domain whose maximal ideal is \(M\) (i.e., for every \(x\in K^*\), \(x\not\in\Rightarrow x^{-1}\in V\)), the prime ideals of \(V\) are the prime ideals of \(R\), and \(R\) is a valuation ring if and only if \(R= V\). Let \(R\subset B\) be an extension of integral domains and \(S\) be a non-empty subset of \(\mathbb{N}^*\), \(R\) is said to be \(S\)-root closed in \(B\) if: \(\forall x\in B\) \((\exists n\in S,\, x^n\in R)\Rightarrow x\in R\), (if \(S= \{n\}\) we say \(n\)-root closed in \(B\)), \(R\) is said to be \(S\)-closed in \(B\) if: \(\forall x\in B\) \((\forall n\in S,\,x^n\in R)\Rightarrow x\in R\). \(R\) is said to be \(F\)-closed in \(B\) if: \(\forall x\in B\) \((\exists n\in S,\,x^2\in R,\, x^3\in R,\, nx\in R)\Rightarrow x\in R\), \(R\) is said to be seminormal in \(B\) if: \(\forall x\in B\) \((x^2\in R,\, x^3\in R)\Rightarrow x\in R\). Assume that \(R\) is a PVD such that there exists a field \(K\) and a subfield \(K_0\) of \(K\) which satisfies: \(R= K_0+ M\), \(V= K+ M\).NEWLINENEWLINE The authors prove that if \(K_0\) is seminormal (resp. \(m,n\)-root closed, \(S\)-root closed, \(S\)-closed) in \(K\), then \(R\) is seminormal (resp. \(mn\)-root closed, \(S\)-root closed, \(S\)-closed) in \(\text{frac}(R)\) (the fraction field of \(R\)). Assume that \(R\subset B\subset C\) is an extension of PVD's, where \(B\) is \(n\)-root closed (resp. seminormal, \(F\)-closed) in \(C\), then they prove that \(R\) is \(n\)-root closed (resp. seminormal, \(F\)-closed) in \(B\) if and only if \(R\) is \(n\)-root closed (resp. seminormal, \(F\)-closed) in \(C\). Assume that \(B\) is a domain extension of a PVD \(R\) and \(P\) is a prime ideal of \(R\) which is also an ideal of \(B\); they prove that if \(R\) is \(n\)-root closed (resp. seminormal, \(S\)-root closed, \(S\)-closed, \(F\)-closed) in \(B\), then \(R/P\) is \(n\)-root closed (resp. seminormal, \(S\)-root closed, \(S\)-closed, \(F\)-closed) in \(B/P\).NEWLINENEWLINE An integral domain is said to be atomic if every non-unit is a product of a finite number of irreducible elements (atoms); furthermore it is a half factorial domain (HFD) if every irreducible elements \(x_1,\dots, x_m\), \(y_1,\dots, y_n\), \(x_1\cdots x_m= y_1\cdots y_n\Rightarrow m= n\). A strongly half-factorial domain (SHFD) is a HFD such that every overring is a HFD, and a locally domain (LHFD) is a HFD such that each of its localizations is a HFD. They prove that every Noetherian PVD is a SHFD and a LHFD. A HFD \(R\) is a boundary valuation domain (BVD) if for every \(\alpha= (x_1\cdots x_m)/(y_1\cdots y_n)\in \text{frac}\) such that \(m\neq n\) (where \(x_1,\dots, x_m\), \(y_1,\dots, y_n\) are atomes), \(\alpha\not\in R\Rightarrow \alpha^{-1}\in R\). The prove that if \(R\) is an integral domain, then \(R\) is a BVD if and only if \(R\) is an atomic PVD.NEWLINENEWLINE Finally, let \(R\subset B\) be a domain extension such that NEWLINE\[NEWLINE\forall b\in B,\;\exists u\in U(B),\;\exists a\in R,\;b= ua\tag{\(*\)}NEWLINE\]NEWLINE (where \(U(B)\) is the group of inversible elements of \(B\)). If \(R\) is a PVD which satisfies \((*)\), then \(B\) is a PVD. Furthermore, if \(\text{Spec}= \text{Spec}(B)\), then \(B\) has no atoms if and only if \(R\) has no atoms. They also provide many examples of rings satisfying those properties and of different cases.