Minimal Ring Extensions Research Articles

Certain towers of ramified minimal ring extensions of commutative rings, II

Let [Formula: see text] be an odd prime number and [Formula: see text]. Assume that either [Formula: see text]i[Formula: see text] [Formula: see text] or [Formula: see text]ii[Formula: see text] [Formula: see text] and [Formula: see text] is congruent to either [Formula: see text] or [Formula: see text] modulo [Formula: see text] [Formula: see text]respectively, assume that [Formula: see text] and [Formula: see text] is congruent to either [Formula: see text] or [Formula: see text] modulo [Formula: see text]. Then there exist exactly five [Formula: see text]respectively, exactly seven[Formula: see text] isomorphism classes of rings [Formula: see text] for which there exists a tower [Formula: see text] of ramified [Formula: see text]integral minimal[Formula: see text] ring extensions such that [Formula: see text] is the only ring properly contained between [Formula: see text] and [Formula: see text]. We produce a set of isomorphism class representatives of such [Formula: see text] and, for each such [Formula: see text], we show that, up to isomorphism, the corresponding ring [Formula: see text] is the idealization [Formula: see text]. One consequence, for each integer [Formula: see text] whose prime-power factorization [Formula: see text] (with pairwise distinct prime numbers [Formula: see text] and positive integers [Formula: see text]) satisfies [Formula: see text] for all [Formula: see text], is a classification, up to isomorphism, of the rings that have cardinality [Formula: see text] and exactly two proper (unital) subrings.

On minimal ring extensions of finite rings

Two conditions, (i) and (ii), are defined, that may hold for a given (unital) ring extension R ⊂ S of (unital, associative, not necessarily commutative) finite rings. It is shown that if S is commutative, then ``"either (i) or (ii)” is a necessary and sufficient condition for R ⊂ S to be a minimal ring extension; and that for such extensions, (i) and (ii) are logically independent. For extensions with S (finite and) noncommutative, "either (i) or (ii)” is neither necessary nor sufficient for R ⊂ S to be a minimal ring extension; and for such minimal ring extensions, (i) and (ii) are logically independent. Next, let R ⊂ Sj be minimal ring extensions with Sj (finite and) commutative (for j=1,2) and R local. Then: S1 and S2 are the same type (that is, ramified, decomposed or inert) of minimal extension of R ↔ |Z(S_1)|=|Z(S_2)| ↔ |U(S_1)|=|U(S_2)|.

On almost valuation ring pairs

If [Formula: see text] are (commutative) rings, [Formula: see text] denotes the set of intermediate rings and [Formula: see text] is called an almost valuation (AV)-ring pair if each element of [Formula: see text] is an AV-ring. Many results on AV-domains and their pairs are generalized to the ring-theoretic setting. Let [Formula: see text] be rings, with [Formula: see text] denoting the integral closure of [Formula: see text] in [Formula: see text]. Then [Formula: see text] is an AV-ring pair if and only if both [Formula: see text] and [Formula: see text] are AV-ring pairs. Characterizations are given for the AV-ring pairs arising from integrally closed (respectively, integral; respectively, minimal) ring extensions [Formula: see text]. If [Formula: see text] is an AV-ring pair, then [Formula: see text] is a P-extension. The AV-ring pairs [Formula: see text] arising from root extensions are studied extensively. Transfer results for the “AV-ring” property are obtained for pullbacks of [Formula: see text] type, with applications to pseudo-valuation domains, integral minimal ring extensions, and integrally closed maximal non-AV subrings. Several sufficient conditions are given for [Formula: see text] being an AV-ring pair to entail that [Formula: see text] is an overring of [Formula: see text], but there exist domain-theoretic counter-examples to such a conclusion in general. If [Formula: see text] is an AV-ring pair and [Formula: see text] satisfies FCP, then each intermediate ring either contains or is contained in [Formula: see text]. While all AV-rings are quasi-local going-down rings, examples in positive characteristic show that an AV-domain need not be a divided domain or a universally going-down domain.

On the nature and number of isomorphism classes of the minimal ring extensions of a finite commutative ring

Let (A, M) be any finite local (nonzero commutative unital) ring. Then the set of A-algebra isomorphism classes represented by (equivalently, consisting of) rings B such that is a decomposed (minimal ring) extension is in one-to-one correspondence with the collection of sets of ideals of A such that (internal direct sum), with any such corresponding to the class represented by the ring If A is not a field, then the (finite) set of A-algebra isomorphism classes represented by (equivalently, consisting of) rings B such that is a ramified (minimal ring) extension has cardinality at least 2; known examples show that this bound cannot be improved. If X denotes an indeterminate over the field then is of minimal cardinality in the class of finite local rings A such that A is not a field and the A-algebra isomorphism classes represented by a ramified extension of A constitute fewer than half of the (finitely many) A-algebra isomorphism classes represented by a minimal ring extension of A.

Where Some Inert Minimal Ring Extensions of a Commutative Ring Come from

Where Some Inert Minimal Ring Extensions of a Commutative Ring Come from

Normal pairs of noncommutative rings

This paper extends the concept of a normal pair from commutative ring theory to the context of a pair of (associative unital) rings. This is done by using the notion of integrality introduced by Atterton. It is shown that if $$R \subseteq S$$ are rings and $$D=(d_{ij})$$ is an $$n\times n$$ matrix with entries in S, then D is integral (in the sense of Atterton) over the full ring of $$n\times n$$ matrices with entries in R if and only if each $$d_{ij}$$ is integral over R. If $$R \subseteq S$$ are rings with corresponding full rings of $$n\times n$$ matrices $$R_n$$ and $$S_n$$, then $$(R_n,S_n)$$ is a normal pair if and only if (R, S) is a normal pair. Examples are given of a pair $$(\Lambda , \Gamma )$$ of noncommutative (in fact, full matrix) rings such that $$\Lambda \subset \Gamma $$ is (resp., is not) a minimal ring extension; it can be further arranged that $$(\Lambda , \Gamma )$$ is a normal pair or that $$\Lambda \subset \Gamma $$ is an integral extension.

Characterizing finite fields via minimal ring extensions

Let A be a finite local (commutative unital) ring. If A is not a field, there exists a positive integer N such that for each ring B such that is an inert (minimal ring) extension. It follows that A is a (finite) field the inert extensions of A form infinitely many (equivalently, denumerably many) A-algebra isomorphism classes the minimal ring extensions of A form infinitely many (equivalently, denumerably many) A-algebra isomorphism classes there exists a minimal ring extension such that B is a flat A-module there exists a minimal ring extension such that B is a local ring and is a Galois ring extension.

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

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.

Certain towers of ramified minimal ring extensions of commutative rings

ABSTRACTLet p be a prime number. Let B be the idealization ℤ(+)ℤ∕pℤ and N: = pℤ(+)ℤ∕pℤ. Let B⊂R be a ramified (integral minimal) ring extension, with crucial maximal ideal 𝒩 and (necessarily) an element y such that R = B[y], y2∈B, y3∈B and y𝒩⊆𝒩. Then B is the only (commutative unital) ring properly contained between ℤ and R if and only if 𝒩 = N and either or py∉ℤ. Moreover, there are exactly three isomorphism classes of such rings R. Let A: = ℤ∕pαℤ for some integer α≥2. Let A⊂D⊂E be a tower of ramified ring extensions, (necessarily) with an element z such that E = D[z], z2∈D, z3∈D and zℳ⊆ℳ, where ℳ is the crucial maximal ideal of D⊂E. Then D is the only ring properly contained between A and E if and only if either z2∉A or pz∉A. There are at least two, but only finitely many, isomorphism classes of such rings E. We complete the characterization of the rings with exactly two proper (unital) subrings, including a classification up to isomorphism of these rings in the case of characteristic 0.

PROPERTIES OF RING EXTENSIONS INVARIANT UNDER GROUP ACTION

Let G be a subgroup of the automorphism group of a commutative ring with identity T. Let R be a subring of T. We show that RG ⊂ T G is a minimal ring extension whenever R ⊂ T is a minimal extension under various assumptions. Of the two types of minimal ring extensions, integral and integrally closed, both of these properties are passed from R ⊂ T to RG ⊆ T G. An integrally closed minimal ring extension is a flat epimorphic extension as well as a normal pair. We show that each of these properties also pass from R ⊂ T to RG ⊆ T G under certain group action.

On Finite Maximal Chains of Weak Baer Going-Down Rings

Some recent results of Ayache on going-down domains and extensions of domains that either are residually algebraic or have DCC on intermediate rings are generalized to the context of extensions of commutative rings. Given a finite maximal chain 𝒞 of R-subalgebras of a weak Baer ring T, it is shown how a “min morphism” hypothesis can be used to transfer the “going-down ring” property from R to each member of 𝒞. The integral minimal ring extensions which are min morphisms are classified. The ring extensions satisfying FCP (i.e., for which each chain of intermediate rings is finite) are characterized as the strongly affine extensions with DCC on intermediate rings. In the relatively integrally closed case, such extensions R ⊆ T induce open immersions Spec(S) → Spec(R) for each R-subalgebra S of T.

On Minimal Extensions of Rings

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.

ON MAXIMAL NON-ACCP SUBRINGS

A domain R is a maximal non-ACCP subring of its quotient field if and only if R is either a two-dimensional valuation domain with a DVR overring or a one-dimensional nondiscrete valuation domain. If R ⊂ S is a minimal ring extension and S is a domain, then (R,S) is a residually algebraic pair. If S is a domain but not a field, a maximal non-ACCP subring extension R ⊂ S is a minimal ring extension if (R,S) is a residually algebraic pair and R is quasilocal. Results with a similar flavor are given for domains R ⊂ S sharing a nonzero ideal, with applications to rings R of the form A + XB[X] or A + XB[[X]]. If R ⊂ S is a minimal ring extension such that R is a domain and S is not (R-algebra isomorphic to) an overring of R, then R satisfies ACCP if and only if S satisfies ACCP.

A Sufficient Condition for a Minimal Ring Extension to Be an Overring

It is proved that if R ⊂ T is a minimal ring extension such that (a) the set of zero-divisors of R is contained in a nonmaximal prime ideal of R and (b) each non-zero-divisor of R remains a non-zero-divisor in T, then T is isomorphic to an R-subalgebra of the total quotient ring of R. A nondomain example R ⊂ T satisfying the above hypotheses is given where the Krull dimension of R is any preassigned integer n ≥ 2 and the total quotient ring of R has Krull dimension n − 1. Examples also show that neither of the hypothesis (a), (b) can be deleted. In the case that T is a domain, the above result recovers a theorem of Sato et al. (1992).

Every Commutative Ring Has a Minimal Ring Extension

Let R be a commutative unital ring and E a unital R-module. Then the canonical injective ring homomorphism from R into the idealization R(+) E is a minimal ring homomorphism if and only if E is a simple R-module. For E nonzero, R(+)E is not (R-algebra isomorphic to) an overring of R. If E 1 and E 2 are nonisomorphic simple R-modules, then R(+) E 1 and R(+) E 2 give minimal ring extensions of R which are not isomorphic as R-algebras. The ring of dual numbers over R is a minimal ring extension of R ⇔ R × R is a minimal ring extension of R ⇔ R is a field.

A classification of the minimal ring extensions of an integral domain

Let R be any integral domain. The minimal (commutative unital) ring extensions S of R are, up to R-algebra isomorphism, of three nonoverlapping types: (i) the domains S that contain R and are minimal ring extensions of R; (ii) the idealizations R ( + ) R / M arising from maximal ideals M of R; and (iii) the direct products R × R / M arising from maximal ideals M of R. Distinct maximal ideals of R lead to nonisomorphic idealizations (respectively direct products) in case (ii) (respectively case (iii)). If R is not a field, then distinct domains S arising in case (i) within the same quotient field of R are not isomorphic as R-algebras. If R is a field K, then the domains S arising in (i) are the minimal (necessarily algebraic) field extensions of K; in this case, distinct such fields S 1 , S 2 are K-algebra isomorphic if and only if S i = K ( α i ) where α 1 , α 2 are roots of the same irreducible polynomial in K [ X ] .

A classification of the minimal ring extensions of certain commutative rings

All rings considered are commutative with identity and all ring extensions are unital. Let R be a ring with total quotient ring T. The integral minimal ring extensions of R are catalogued via generator-and-relations. If T is von Neumann regular and no maximal ideal of R is a minimal prime ideal of R, the minimal ring extensions of R are classified, up to R-algebra isomorphism, as the minimal overrings (within T) of R and, for maximal ideals M of R, the idealizations R ( + ) R / M and the direct products R × R / M . If T is von Neumann regular, the minimal ring extensions of R in which R is integrally closed are characterized as certain overrings, up to R-algebra isomorphism, in terms of Kaplansky transforms and divided prime ideals, generalizing work of Ayache on integrally closed domains; no restriction on T is needed if R is quasilocal. One application generalizes a recently announced result of Picavet and Picavet-L'Hermitte on the minimal overrings of a local Noetherian ring. Examples are given to indicate sharpness of the results.

On the FIP Property for Extensions of Commutative Rings #

ABSTRACT A (unital) extension R ⊆ T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ⊂ S ⊂ T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ⊂ T is a (module-) finite minimal ring extension, then R(X)⊂T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ⊆ T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ⊆ R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ⊆ R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ℤ which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (ℤ:R)≠0. Some results are also given for the residually FIP property.