Pairs of Rings with the Same Prime Ideals

Abstract

There are numerous instances in which the partners in an extension of commutative rings R ⊂ T have the same prime ideals, i.e., in which Spec(R) = Spec(T). Although this equality is intended to be taken set-theoretically, the identification easily extends to the corresponding spaces endowed with their Zariski topologies (see Proposition 3.5(a)), but of course need not extend to an identification of Spec(R) and Spec(T) as affine schemes. Perhaps the most striking recent illustration of this phenomenon arises from the work of Hedstrom and Houston [14] in which R is a pseudo-valuation domain and T is a suitable valuation overring. Other examples may be found by means of the D + M construction, either in its traditional form [12, p. 560] or in the generalized situation introduced by Brewer and Rutter [5].

Note on the divisoriality of domains of the form $k[[X^{p}, X^{q}]]$, $k[X^{p}, X^{q}]$, $k[[X^{p}, X^{q}, X^{r}]]$, and $k[X^{p}, X^{q}, X^{r}]$

“…(see [28] [1,2,3,4,5,6,7,8,9,13,14,15,16,17,21,20,22,25,26,29] [24]. A one-dimensional domain is Gorenstein if and only if the inverse of the maximal ideal is generated by two elements [10].…”

Some Applications of André–Quillen Homology to Classes of Arithmetic Rings

“…According to Anderson and Dobbs (1980), a quasi-local domain (D, m) with quotient field K is called a pseudo-valuation domain (PVD), if x, y ∈ K and xy ∈ m imply x ∈ m or y ∈ m. Clearly, a valuation domain is a PVD. Next, we characterize the ⋆ PVDs (for other equivalent assertions see Zafrullah (1987, Theorem 4.5)).…”

When is each proper overring of $R$ an S(Eidenberg)-domain?

“…Let us recall some terminology: Let T be a ring, I an ideal of T , D be a subring of T/I and let R be the subring of T defined by the following pullback construction: constructions were considered for the first time in [7], in the contest of general pullback construction. Particularly the last construction to be noted here concerns the notion of a pseudo-valuation domain (for short, a PVD), which was introduced by J. R. Hedstrom and E. G. Houston [9] and has been studied subsequently in [2], [5], [6] and [10]. A domain R is said to be a PVD in case each prime ideal p of R is strongly prime, in the sense that whenever x, y ∈ qf(R) satisfy xy ∈ p, then either x ∈ p or y ∈ p, equivalently, in case R has a (uniquely determined) valuation overring V such that Spec(R) = Spec(V ) as sets, equivalently (by [2, Proposition 2.6]) in case R is a pullback of the form V × K k, where V is a valuation domain with residue field K and k is a subfield of K. As the terminology suggests, any valuation domain is a PVD [9, Proposition 1.1].…”