Geodesic
Pairs of Rings with the Same Prime Ideals, II
Canadian journal of mathematics, Tome 40 (1988) no. 6, pp. 1399-1409

Voir la notice de l'article provenant de la source Cambridge University Press

Much of [2] was devoted to studying pairs of subrings A ⊂ B of a field with the property that A and B have the same prime ideals. In this paper, we continue that investigation, but we no longer assume that A and B are comparable. Interestingly, most of the results of [2] carry over to this more general context. Besides such extensions of [2], additional motivation for the more general context comes from the need to explicate some naturally occurring examples (see Examples 2.5, 3.6, and 4.3).Section 2 begins by showing that we may reduce to the case in which R is a quasilocal domain with nonzero maximal ideal M and quotient field K. Proposition 2.3 establishes that the set C(R) of all subrings A of K with Spec(A) = Spec(R) forms a complete semilattice. 
Anderson, David F.; Dobbs, David E. Pairs of Rings with the Same Prime Ideals, II. Canadian journal of mathematics, Tome 40 (1988) no. 6, pp. 1399-1409. doi: 10.4153/CJM-1988-063-8
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