DOC PREVIEW
MINIMAL SETS IN FINITE RINGS

This preview shows page 1-2-3-24-25-26 out of 26 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 26 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 26 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 26 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 26 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 26 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 26 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 26 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

MINIMAL SETS IN FINITE RINGSLEANNE CONAWAY AND KEITH A. KEARNESAbstract. We describe how to calculate the hα, βi-minimal sets in any finite ring.1. IntroductionIn the mid-1980’s, David Hobby and Ralph McKenzie introduced a general local-ization theory for algebras, called tame congruence theory. The theory has provento be an effective tool for analyzing the structure of finite algebras and locally finitevarieties. A preliminary version of the theory appears in [13], the handbook of thetheory is [4], and later expositions of the theory can be found in [1, 2, 5, 8, 9, 10].One of the strengths of tame congruence theory is that its concepts are language-independent. But this feature raises a question about how the theory might makeconnections with classical algebra. Since the theory studies finite algebras via thestructure and distribution of their minimal sets, this question might be phrased morespecifically as: “How do you calculate the minimal sets in a finite group, ring, module,semigroup, etc?” One of the problems posed by Hobby and McKenzie is:Problem 15 of [4]. Investigate h0, αi-minimal sets for abelian minimal congruencesα of finite groups. Do the same for finite rings.In this paper we solve the second half of Problem 15 by describing all minimal setsin finite rings1up to polynomial isomorphism. As part of the solution we find itnecessary to describe the minimal sets in finite modules and bimodules. The mainresults are Theorems 2.9, 3.5, 4.1, and Corollary 4.2.The part of Problem 15 concerning finite groups seems harder than the part aboutrings, and it is still open. A partial result in this direction was proved in 1996 byK. Kearnes, E. W. Kiss and C. Szabo: if G is a finite group, then any p-Sylowsubgroup of G is a “neighborhood”. (See the next section for this definition. See [8]for a proof of this statement about groups.) This result solves the part of Problem 15that concerns nilpotent groups.1991 Mathematics Subject Classification. Primary 08B05, Secondary 16P10.Key words and phrases. finite ring, minimal set, tame congruence theory.This material is based upon work supported by the National Science Foundation under GrantNo. DMS 9802922.1In this paper, the word “ring” always means “associative ring with unit element”.12 LEANNE CONAWAY AND KEITH A. KEARNESSince the part of Problem 15 that concerns groups is still mostly open, it goeswithout saying that the corresponding problems for semigroups and loops have notbeen solved. However, the E-minimal semigroups and loops have been characterizedin [16, 7] respectively. Identifying the E-minimal algebras in a given variety is a spe-cial but important subcase of the problem of describing all minimal sets in membersof the variety. The classification of E-minimal (nonunital) rings is an unpublishedresult of S. Seif.2. PreliminariesThe following definition is a slight specialization of a concept from [8].Definition 2.1. Let A = hA; F i be an algebra. A neighborhood of A is a subset of Aof the form U = ε(A) where ε is a nonconstant idempotent unary polynomial of A.Neighborhoods U and V are polynomially isomorphic if there exist unary polynomialsf and g of A such that f |U: U → V and g|V: V → U are inverse bijections. Thepolynomials f and g are called polynomial isomorphisms.Note that if A is a finite algebra, U, V ⊆ A are neighborhoods, and there existunary polynomials f and g of A such that f|U: U → V and g|V: V → U areinjective, then U and V must be polynomially isomorphic. The reason for this isthat the finiteness of U and V imply that f|Uand g|Vare bijections. To exhibit apair of inverse bijections, choose n > 0 so that (f ◦ g)n(x) = x on V . Then f|Uand g ◦ (f ◦ g)2n−1|Vare restrictions of unary polynomials and are inverse bijectionsbetween U and V .Let A be an algebra, ε be an idempotent polynomial, and U = ε(A) be the imageof ε. Suppose that f and g are unary polynomials of A such that g ◦ f(x) = x onU. Then f ◦ ε ◦ g is an idempotent polynomial of A. Moreover, if V = f ◦ ε ◦ g(A),then f|U: U → V and g|V: V → U are inverse polynomial bijections, so V is aneighborhood polynomially isomorphic to U.Later in this paper we will focus on algebras that have an underlying additive groupstructure (rings, modules and bimodules). Since the group of additive translations(i.e., the polynomials of the form π(x) = x + a) acts transitively on any such algebra,the observation of the previous paragraph implies that each neighborhood of a ring,module, or bimodule is polynomially isomorphic to a neighborhood of zero.Neighborhoods support algebras that locally approximate the polynomial structureof A. These algebras are called “induced algebras”.Definition 2.2. If A is an algebra and U ⊆ A is a neighborhood, then the (nonin-dexed) algebra that A induces on U isA|U= hU; Pol(A)|Ui.(Here Pol(A)|Umeans the clone on U consisting of the restrictions to U of all poly-nomial operations of A that can be restricted to U.)MINIMAL SETS IN FINITE RINGS 3It is not hard to show that if U and V are polynomially isomorphic neighborhoods,then A|Uand A|Vare isomorphic nonindexed algebras. (See pages 28 and 29 of [4].)Lemma 2.3. Let A be a finite algebra, let U ⊆ A be a neighborhood, and let V ⊆ Ube a subset. Then(1) V is a neighborhood of A iff it is a neighborhood of A|U.(2) If the equivalent conditions in (1) hold, then A|V= (A|U)|V.Proof. For (1), if ε is an idempotent unary polynomial of A whose image is V , thenε|Uis an idempotent unary polynomial of A|Uwhose range is V . Conversely, if ε isan idempotent polynomial of A whose image is U, and ϕ is a polynomial of A thatcan be restricted to U for which ϕ|Uis an idempotent polynomial of A|Uwith imageV , then ϕ ◦ ε is an idempotent polynomial of A whose image is V .Part (2) follows from Exercise 2.5 (2) of [4]. The next result explains the connection between neighborhoods and hα, βi-minimalsets for finite algebras with a Maltsev polynomial.Lemma 2.4. Let A be a finite algebra with a Maltsev polynomial. A subset U ⊆ A isan hα, βi-minimal set for some prime quotient hα, βi of A if and only if U is minimalunder inclusion among neighborhoods of A. If U is minimal under inclusion amongneighborhoods, then U is hα, βi-minimal for any α ≺ β for which α|U6= β|U.Proof. For the forward direction of the first statement, Lemma 4.17 and Theo-rems 4.31 and 8.5 of [4] imply that if A has a Maltsev polynomial and U is anhα,


MINIMAL SETS IN FINITE RINGS

Download MINIMAL SETS IN FINITE RINGS
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view MINIMAL SETS IN FINITE RINGS and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view MINIMAL SETS IN FINITE RINGS 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?