On Minimal Extensions of Rings

Abstract

Given two rings R ⊆ S, S is said to be a minimal ring extension of R if R is a maximal subring of S. In this article, we study minimal extensions of an arbitrary ring R, with particular focus on those possessing nonzero ideals that intersect R trivially. We will also classify the minimal ring extensions of prime rings, generalizing results of Dobbs, Dobbs & Shapiro, and Ferrand & Olivier on commutative minimal extensions. * 2000 Mathematics Subject Classification numbers: 16S70 (primary), 16S20, 16N60(secondary).

Torsional oscillators and the entropy dilemma of putative supersolid4He

“…In Ref. [16] we assumed that fit parameters A and B are independent, however, they obey the constraint A/B ≃ 4π/f 0 [17,18]. This leads to slightly degraded fits of both period and dissipation compared to the unconstrained fits in Ref.…”

On minimal ring extensions of finite rings

“…Note that q is a prime number because we have assumed that the partners in the minimal field extension F ⊂ k are finite fields. In the proof of [17,Lemma 6.6] (where it had only been assumed that F ⊂ k is an n-dimensional field extension with 2 ≤ n < ∞), S was produced as follows: as k as an n-dimensional vector space over F , view R := k as a subring of Hom F (k, k) via the left regular action of k on itself; and take S to be minimal among subrings of Hom F (k, k) ( ∼ = M n (F )) that properly contain k. It was shown in [17,Lemma 6.6] that S ∼ = M n (D) for some (in our case, necessarily finite) division ring D. By a celebrated theorem of Wedderburn (cf. [19, Theorem 3.1.1 and item (2), page 102]), any finite division ring is a field, and so D is a finite field.…”

“…Example 2.6 makes use of an example of George Bergman, as presented and interpreted by Dorsey and Mesyan in [17,Lemma 6.6 and Remark 6.7]. One should note that Bergman's construction is more general than the use to which it is put in Example 2.6, as Bergman's construction did not require the fields F and k in Example 2.6 to be finite (but he did require that F ⊂ k be a finitedimensional field extension).…”

“…As usual, |S| denotes the cardinal number of a set S; F j denotes the finite field with cardinality j; ⊂ denotes proper inclusion; and X denotes an indeterminate over the ambient ring(s). For rings R ⊆ S, [R, S] denotes the set of intermediate rings, that is {T | T is a ring and R ⊆ T ⊆ S}; and, following [18], we say that R ⊂ S is a minimal ring extension if |[R, S]| = 2; that is, if R ⊂ S and there does not exists a ring T such that R ⊂ T ⊂ S. The definition of a minimal ring extension in [18] stipulated that the rings R and S are commutative, but this concept has been extended and studied for arbitrary (that is, possibly noncommutative) rings in recent years, in papers such as [17] and [1].…”

“…For the context of ring extensions R ⊂ S with R noncommutative (and S finite), we find: in Example 2.9 (in the spirit of Example 2.6), by using a matricial result of Al-Kuleab and Jarboui [1], a minimal ring extension satisfying both (i) and (ii) and another minimal ring extension satisfying (i) but not (ii); in Example 2.10 (in the spirit of Example 2.8 (d)), by using upper triangular matrices, a minimal ring extension satisfying neither (i) nor (ii); and in Example 2.11 (in the spirit of Example 2.7), by using Dorroh extensions (in the sense of [17]), a minimal ring extension satisfying (ii) but not (i). Section 2 also contains a number of other analogous examples and a variety of remarks.…”