Going Down: Ascent/Descent

Characterizations of the Integral Domains Whose Overrings Are Going-Down Domains

“…Now, let T be any overring of R. Then T is an overring of R , and so T is a going-down domain. As T ⊆ T satisfies going-down, [32,Lemma B] gives that T must be a going-down domain.…”

“…(iv) ⇒ (v): Assume (iv). By[32, Lemma D], it suffices to prove that if Q ∈ Spec(R ), with n := ht R (Q) (< ∞), and q := Q ∩ R (∈ Max(R)), with m := ht R (q) (< ∞), then n = m. In view of the conditions in (iv), we can assume, without loss of generality, that Q is a non-maximal prime ideal of R and that q is unibranched in R . As the assertion is trivial if Q (or q) is (0), we can also assume that n = 0 and m = 0.…”

“…AYACHE AND D. E. DOBBS As in [13] and [21], R is said to be a going-down domain if the extension R ⊆ T satisfies the going-down property for each domain T that contains R (as a subring); cf. also [32], [33], [15], [22], [17], [31]. By [21,Theorem 1], the rings T that need to be tested (to check that R ⊆ T satisfies the going-down property) may be restricted to be either valuation overrings of R or simple overrings of R. The most natural examples of going-down domains are arbitrary Prüfer domains and domains of (Krull) dimension at most 1.…”

Ascending the divided and going-down properties by absolute flatness

“…Let R be a universally going-down weak Baer ring and an absolutely flat ring morphism f : R → S. Then S is a universally going-down weak Baer ring. McAdam proved a result (see [23]) that gives in our context an ascent criterion: if R → S is an absolutely flat ring morphism between weak Baer rings and R is a going-down ring, then S is a going-down ring if and only if S is treed. We deduce from this criterion the next result.…”