Ring extensions of length two

Abstract

We characterize extensions of commutative rings R ⊂ S such that R ⊂ T is minimal for each R-subalgebra T of S with T = R, S. This property is equivalent to R ⊂ S has length 2. Such extensions are either pointwise minimal or simple. We are able to compute the number of subextensions of R ⊂ S. Besides commutative algebra considerations, our main result is a consequence of the recently introduced by van Hoeij et al. concept of principal subfields of a finite separable field extension. As a corollary of this paper, we get that simple extensions of length 2 have FIP.2010 Mathematics Subject Classification. Primary:13B02,13B21, 13B22, 12F10; Secondary: 13B30.Key words and phrases. FIP, FCP extension, minimal extension, length of an extension, integral extension, support of a module, t-closure, algebraic field extension, separable field extension, principal subfield, pointwise minimal extension.

Boolean FIP ring extensions

“…In fact, Theorem 3.10 says that k ⊂ L is Boolean if and only if any T ∈ [k, L[ is an intersection in a unique way of finitely many co-atoms. Thanks to principal subfields introduced in [35] and some of their properties we studied in [31], we are able to characterize co-atoms of a finite separable field extension, and then give a characterization of a finite separable Boolean field extension by using [35], from van Hoeij, Klüners and Novocin, that gives an algorithm to compute subextensions of a finite separable field extension. We recall the notation of [31], (k u [X] is the set of monic polynomials of k[X]).…”

“…The L ′ α s are called the principal subfields of k ⊂ L. It may be that L α = L β for some α = β (see [31,Example 5.17 (1)]). To get rid of this situation, we defined in [31]…”

“…For α ∈ N r , we have g α (X) = (X − x)f α (X) ∈ D, giving that there exists some K ∈ [k, L] such that g α = f K with L = K because f L (X) = X − x. It follows that K ⊂ L is minimal by [31,Lemma 5.7]. The monic minimal polynomial of x over k is f (x) := X 6 − 2 = (X −x)(X +x)(X 2 +xX +x 2 )(X 2 −xX +x 2 ), which is its decomposition into irreducible polynomials over L. Set f 1 (X) := X + x, f 2 (X) := X 2 + xX + x 2 , f 3 (X) := X 2 − xX + x 2 and g α (X) := (X − x)f α (X), for α = 1, 2, 3.…”

“…In a recent paper [31], we characterized ring extensions R ⊂ S of length 2 and gave the value of |[R, S]|. It is then easy to characterize an extension of length 2 which is Boolean.…”

Quasi-Prüfer Extensions of Rings

“…Since [42,Proposition 4.18(2)] asserts that M ′ ∈ MSupp(R(X)/R(X)) = MSupp(R(X)/R(X)), this gives that M ∈ MSupp(R/R), which entails that M ∈ MSupp(S/R). By [43,Lemma 1.8], there exists…”