t-closed pairs (Q2785949)

A \(t\)-closed (seminormal) pair is a pair \((A,B)\) of rings such that \(A\subset B\) and every ring \(C\) between \(A\) and \(B\) is \(t\)-closed (seminormal). Here \(t\)-closed means that whenever \(x^3+rxy+y^2=0\) with \(x,y,r\in C\) there exists \(z\in C\) with \(x=z^2-rz\), \(y=z^3-rz^2\). Obviously every \(t\)-closed ring is seminormal (i.e. satisfies the condition for \(r=0\)). NEWLINENEWLINENEWLINEThe main result is the following: Let \(A\to B\) be an injective morphism of weak Baer ring with \(\text{Bool}(A)= \text{Bool}(B)\) such that \((A,B)\) is a seminormal pair; then it is \(t\)-closed if and only if the induced map \(\text{Spec}(B)\to \text{Spec}(A)\) is bijective. (A ring is a weak Baer ring, if the annihilator of any principal ideal is generated by an idempotent, and \(\text{Bool}(A)\) is the set of idempotents.) -- Furthermore, it is shown that a 1-dimensional integral domain \(A\) whose normalization is a Prüfer domain is \(t\)-closed if and only if it is locally a pseudo-valuation domain (i.e. has a valuation overring with the same spectrum). Finally the author studies the condition for an integral domain to have all its underrings \(t\)-closed: This is the case if and only if \(A\) is the ring of integers or an algebraic extension of a finite field.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].