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].