A minimal ring extension of a large finite local prime ring is probably ramified

Abstract

Given any minimal ring extension [Formula: see text] of finite fields, several families of examples are constructed of a finite local (commutative unital) ring [Formula: see text] which is not a field, with a (necessarily finite) inert (minimal ring) extension [Formula: see text] (so that [Formula: see text] is a separable [Formula: see text]-algebra), such that [Formula: see text] is not a Galois extension and the residue field of [Formula: see text] (respectively, [Formula: see text]) is [Formula: see text] (respectively, [Formula: see text]). These results refute an assertion of G. Ganske and McDonald stating that if [Formula: see text] are finite local rings such that [Formula: see text] is a separable [Formula: see text]-algebra, then [Formula: see text] is a Galois ring extension. We identify the homological error in the published proof of that assertion. Let [Formula: see text] be a finite special principal ideal ring (SPIR), but not a field, such that [Formula: see text] has index of nilpotency [Formula: see text] ([Formula: see text]). Impose the uniform distribution on the (finite) set of ([Formula: see text]-algebra) isomorphism classes of the minimal ring extensions of [Formula: see text]. If [Formula: see text] (for instance, if [Formula: see text]), the probability that a random isomorphism class consists of ramified extensions of [Formula: see text] is at least [Formula: see text]; if [Formula: see text] (for instance, if [Formula: see text] for some odd prime [Formula: see text]), the corresponding probability is at least [Formula: see text]. Additional applications, examples and historical remarks are given.

Characterizing finite Boolean rings by using finite chains of subrings

“…Next, for the induction step, we can assume that 2 ≤ k < m and we have arranged that D k is T k+1 . Since D k ⊂ D k+1 is a minimal ring extension, [7,Lemma 2.2] shows that for some (uniquely determined) index i such that 1…”

“…This observation has been of enormous use in studying minimal ring extensions that involve (especially finite) commutative rings, because a commutative ring extension of a finite direct product of n nonzero rings is isomorphic to a direct product of commutative ring extensions of the given n direct factors (cf. [7,Lemma 2.2]). Unfortunately, no such description of minimal ring extensions is possible for noncommutative rings.…”

“…Iterate the process, ending with an augmentation with a chain whose i th member is obtained by replacing what had been the d th direct factor (equal to F p ) of T d with the i th member of C d . As before, each newly added step is inert, by [7,Lemma 2.2]. The upshot is a finite maximal chain of rings, say C, going from F p to d j=1 K j = R, whose first d − 1 steps are decomposed and whose subsequent steps are all inert.…”

“…In Example 2.5, B 2 is neither a B 1 -algebra nor a nontrivial direct product. The latter aspect means that one cannot study such data by using some methods that have been fruitful in dealing with finite chains of commutative ring extensions, such as those summarized in [7,Lemma 2.2]. To avail ourselves of those methods here, we introduce the "Adjusted FMC" condition (in short, AFMC) that restricts attention to the finite maximal increasing chain of rings with the property that their last step…”

On minimal ring extensions of finite rings

“…[2,Theorem 8.7]), but such a requirement would be too restrictive to accommodate a broad array of noncommutative ring extensions. So, prior to the statement of Lemma 2.4, we reformulate conditions (i) and (ii) (which had appeared in the statement of Theorem 2.1) so that their new definitions are not connected to any supposed direct product descriptions of R and S. (For finite commutative rings R ⊂ S, these reformulations have no effect if R is local, while if R is not local, these reformulations have an insignificant effect, in view of [9,Lemma 2.2].) For commutative rings, Theorem 2.1 (together with the comments immediately following its proof) shows that that (i) and (ii) are logically independent (regardless of whether one considers their original versions or their reformulated versions).…”