Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
ATZ-Fachkongress

Chassis.tech plus 2024


4 Kongresse in einer Veranstaltung

Treffen Sie in München die Fachleute von Chassis, Lenkung, Bremsen sowie Reifen/Räder.
Erweiterte Suche
Anmelden
nach oben

2002 | Buch

Elements and Complements Kapitel lesen Erstes Kapitel lesen

Nearrings

Some Developments Linked to Semigroups and Groups

verfasst von: Celestina Cotti Ferrero, Giovanni Ferrero

Verlag: Springer US

Buchreihe : Advances in Mathematics

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

This work presents new and old constructions of nearrings. Links between properties of the multiplicative of nearrings (as regularity conditions and identities) and the structure of nearrings are studied. Primality and minimality properties of ideals are collected. Some types of `simpler' nearrings are examined. Some nearrings of maps on a group are reviewed and linked with group-theoretical and geometrical questions.
Audience: Researchers working in nearring theory, group theory, semigroup theory, designs, and translation planes. Some of the material will be accessible to graduate students.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Elements and Complements
Abstract
We give a first overview on the subject stating more frequently used definitions, elementary results, more exciting examples and celebrated results that also suggest to the reader to view the chapters that follows.
We introduce the more important class of examples of nearrings: they are mainly particular nearrings of functions from an additive group G to itself, equipped with pointwise sum and composition product as, for example, are the nearrings of zero-preserving maps and the nearrings generated by endomorphisms, automorphisms and inner automorphisms of G (the latter are distributively generated). Centralizer nearrings are usually just the subnearrings of previous nearrings obtained taking only the elements commuting with all elements of a particularly interesting semigroup of functions.
Also, nearrings of polynomials (with composition product or with pletysm) on a ring and polynomials on a group are introduced.
Affine nearrings and trivial nearrings (and nearfields) are introduced not only as historical note, but also because they are useful to realize how it is possible to work in the nearring theory.
An initial study on planar, distributively generated and local nearrings will be useful for when we will encounter these structures in following chapters. We also will introduce group- and matrix nearrings, generated by the dream to generalize classical representation theory of groups to representations of a group (or even a semigroup) as a group (or semigroup) of automorphisms (or endomorphisms) of a non-abelian group.
Some impressive results as, for example, that of the type by Fröhlich (see 1.2.13, 1.2.17, 1.2.18) and density theorems of the type by Jacobson are well known, and so recalled without proofs.
In this chapter (as well as throughout this book) we are particularly interested in giving only results that can be easily proven to accustom the reader to this not-so-common subject and to set tools for the other chapters.
So, we introduce the most used conditions of (maybe localized) distributivity (see 1.7.49), regularity, and introduce the condition IFP and (usual) chain conditions.
Some important elementary tools are also introduced as, for example, Clay functions and “Pierce” and “Fitting” decompositions, but only a touch upon various radicals are given because more used topics already appeared in other books.
Furthermore, in this chapter, as in this book’s entirety, we will give some suggestions for further researches.
Celestina Cotti Ferrero, Giovanni Ferrero
Chapter 2. Constructions
Abstract
As in many mathematical subjects, a characterization of a class of objects is more precise and interesting if it can lead to construing all such objects, in particular, when the obtained constructions can also help to find other properties of the objects or to give a more detailed classification of them. Conversely, a construction itself can help to define new and maybe interesting objects to be further studied (and maybe, construed with a smarter technique).
The first natural idea to construe nearrings is to choose an additive group [G; +] and try to define a product fulfilling required properties, also because many mathematicians are acquainted with elementary group theory. Even more simple is to start from the additive group of a vector space (or even from a vector space V) because, in this case, linear transformations (that are particular endomorphisms of [V; +]) are immediately at our disposal, and can be used as Clay functions. Also, to give an example of using this idea (and to not be too heavy writing down complex conditions), we open the chapter with a construction of an interesting local nearring and (in a very simplified form) of a nearring that forth arouses in a framework that is a bit further away from the usual nearringers’ studies.
We can read as a generalization of the previous ideas some constructions obtained by starting from a direct (or semidirect) sum of additive groups; some of them can be obtained by “composition” of other nearrings using variations of classical compositions of additive groups or of multiplicative semigroups.
As usual, only the simpler remarks are given, to be also used as examples of natural techniques.
The first section of this chapter closes with a more sophisticated construction of a group-theoretical source, pointing out some nearrings fulfilling peculiar properties.
The second section is devoted to the analysis of certain nearrings arising from the theory of automata: they are examples of the idea of define a nearring [N; +, ·] starting from a group [G; +] and defining a product by xy = α(x) + β(y) for suitable functions α, β from G to G.
To go deeper into the study of nearrings with a given additive group [G; +] obtained by gluing “partial” Clay functions obtained using few endomorphisms of G, it is suggested to study generalized orbits of a semigroup of endomorphisms of an additive group. Some applications of this idea are given in construing, for example, nearrings whose Clay semigroup is a given infinite cyclic semigroup with zero added.
This leads to an important theorem (BFKL) which teaches us to construe, under suitable (not so elegant) conditions, nearrings with a given additive group and a given Clay semigroup. This theorem is the key point of the section (and of the chapter) because it is followed by variations and applications of great interest. In fact, perhaps, it was given to find the groups hosting nontrivial nearrings (and it is close to solving the problem), but (with its many variants), it can also be used to construe and classify many other classes of nearrings. Here it is used for a little study on planar and integral nearrings also.
The last section is devoted to (sometimes trivial) various techniques that may be used to obtain nearrings “deforming” operations of other structures (a Dickson’s idea), and also to generalize some of the results previously written as a preparation of the BFKL theorem: after such theorem, we are free to treat certain questions, even if such theorem is not explicitly used. Another idea stressed in this section is the Magill “lamination.” We introduce it, but we avoid the topological considerations from which such idea arose.
Celestina Cotti Ferrero, Giovanni Ferrero
Chapter 3. Regularities and More
Abstract
This chapter is devoted to nearrings N fulfilling regularity conditions (as a ∈ aNa),variants and generalizations: they are obviously semigroup­theoretical conditions arising by classical studies on rings. As usual, we will try to see if (and how) it is possible to bring ring-theoretical results to nearring theory. It is easy to think that there are many variants of this type of idea, and each variant can be very interesting, or not so useful, depending on the context of our study.
We introduce some properties of idempotent elements of a nearring N (with or without an identity u), depending on them belong or do not belong to the multiplicative center of N. We also introduce Pierce decompositions and some lifting of idempotents (recalling classical lift­ing for groups and rings) that are maybe able to suggest semidirect decompositions of particular extensions of the Schreier type, but we do not develop it to remain simple.
Properties of reduced nearrings (if necessary fulfilling other conditions), usually dispersed throughout many papers according to the needs of each paper are collected. In this case other results on idempotents are given.
Several variants of typical regularity conditions are introduced (main­ly in definition 3.3.4) trying to fix a terminology (sometimes creating terms that may be useful to quickly recall important links between such conditions), and warnings on the different terms used by various authors treating this subject.
For example, links between right strongly regular, left strongly regular and regular nearrings are stressed. Obviously, to deepen the study on the structure of regular and strongly regular nearrings we use several ideas from the previous chapters.
Now it is possible to introduce structure theorems for regular or strongly regular nearrings both for the zero-symmetric case and for general cases. For nearrings fulfilling some suitable chain conditions we have sharper structural results.
Particular classes of regular nearrings are introduced as, for example, ortodox nearrings (i.e. nearrings in which the set of idempotents is a semigroup) and as generalized nearfields (i.e. nearrings in which the multiplicative semigroup is inverse: ∀a ∈ N there exists a unique b ∈ N such that a = aba and b = bab). A class of nearrings containing the class of generalized nearfields is the class of biregular nearrings (a nearring N is biregular if for all a ∈ N there exists a central idempotent e such that <a> N = eN). Proving that this class is the class of Betsch’s biregular nearrings we have a structure theorem for such nearrings.
Some variants of nearrings fulfilling regularity conditions for stable (left or right) and right (left) bipotent nearrings (i.e. nearrings N fulfilling conditions as Na = aNa or aN = a 2 N for all a ∈ N) are introduced. An interesting construction of stable nearrings is given using an idea of “semidirect sum” introduced in chapter II.
A tiny study of right bipotent nearrings fulfilling some chain conditions is given.
To end the chapter, we collect various cases in which regularity con­ditions or other semigroup-theoretical conditions on a nearring N force N to be a nearfield.
Celestina Cotti Ferrero, Giovanni Ferrero
Chapter 4. Multiplicative Identities and Commutativity Conditions
Abstract
We now have realized that semigroup theory can be useful for studying nearrings, so it is natural to examine nearrings whose multiplicative semigroup (or whose Clay semigroup) belongs to a given variety of semigroups. To begin (we will always treat only the initial part of various theories because at this point we will avoid delving into the more complex developments), we can study nearrings fulfilling some multiplicative identities.
As a preamble, we will discuss some of the questions (such as, for example, a generalization of a Putcha-Yaqub theorem) on semigroups to suggest the use of certain semigroup-theoretical invariants to classify and to study nearrings fulfilling multiplicative identities.
However, because we are not aware of any other in-depth studies on the general question of seeing when a semigroup is the multiplicative semigroup of a nearring, we will therefore go directly to the study of the nearrings N with certain weak distributivity conditions that imply conditions on the multiplicative semigroup [N;·].
More common studies are on a nearring N fulfilling given identities of small size, mainly in the cases in which N is simple or subdirectly irreducible.
Classical results, by Wedderburn, Jacobson and Herstein, giving sufficient conditions for having commutativity of a ring do not use such identities, but instead, they use less demanding conditions (such as x n ( x ) = x, which is equivalent to ask that each x multiplicatively generates a finite group), usually absorbing the most important identities of small size (as the Boolean identity x 2 = x). So, we will discuss the nearrings fulfilling the more interesting conditions (borrowed from classical papers) collected in definition 4.2.2, to also encounter simpler semigroup identities.
In particular, we will discuss potent and periodic nearrings, also using studies on the condition x—x n (x) ∈ Z (here called Herstein’s condition), and we will meet, as potent, all Boolean nearrings and, as periodic, all near-idempotent nearrings and all self-distributive nearrings. The last cases are simple enough to give simple sharp results. Also, conditions being similar to the classical (xy) n (x,y) = xy are studied (under the name of PC-conditions).
To return to the historical source of these topics, we will have a taste of (multiplicative, but also additive) commutativity results on nearrings. Of course, a complete report cannot be given here because usually ring-theoretical techniques do not work on nearrings and even more complex discussions are necessary to get nontrivial results, so we (after the commutativity results obtained in studying previously quoted conditions) only give some commutativity results that were obtained by classical conditions on commutators and by discussions on derivation and α-derivations.
Celestina Cotti Ferrero, Giovanni Ferrero
Chapter 5. Prime and Minimal Ideals
Abstract
The general theory on various types of primeness for ideals of a nearring is close enough to the ring-theoretical one even if, as usual, each primeness type for rings generates various non equivalent primeness types for nearrings.
The more recent types of primeness, bringing to the notions of equiprime, strongly prime, 2-primal ideals, seem to have particular interest and are introduced in detail. Each of the many primality types (see definition 5.1.1) generates, in the usual natural way, a notion of “prime” radical (as intersection of the ideals with a given type of primality), and such radicals are true radicals according to general points of view in universal algebra. So, we introduce and give a first study on this subject.
In particular, we give fundamental properties for the equiprime and the 2-primal cases. Maybe it is a surprise to see that, as it happens in commutative rings, an ideal of a 2-prime nearring is 0-prime if and only if it is c-prime.
Useful links among various types of primeness, and among primitivity and primeness are given, and cases in which certain types of primeness are equivalent are collected.
A general definition of regularity (for elements of a nearring), as introduced in [Grönewald and Olivier, 1997], is able to unify the regularity conditions studied in the third chapter, to connect various notions of regularity and primeness for giving very general results. So we introduce such notions also by giving some examples of theire use in simple applications.
It is almost mandatory to report the cases in which the famous Nöther theorem on primary decomposition of an ideal is generalized. We try to accomplish this task in a manner to also give interesting structure theorems for biregular nearrings (see definition 3.3.4 as introduced in the third chapter, where we proved simple properties on the lattice of the ideals).
Studies on the heart of a subdirectly irreducible nearring are linked to the characterization of the minimal ideals of a nearring, and we conclude by touching on touching such questions even if we are forced to set some heavy conditions for obtaining results similar to the ring-theoretical ones.
Celestina Cotti Ferrero, Giovanni Ferrero
Chapter 6. Classes of “Simpler” Nearrings
Abstract
In this chapter, we collect some problems such that the study of them brings us, in a very natural way, to use theorem 2.2.23 and its variations.
Maybe (in contrast with the well-known situation for rings), there are not any nontrivial groups hosting (as additive groups) only trivial nearrings. Today the question is still not close completely, but simple applications of some constructions of the second chapter give very wide classes of groups hosting nontrivial nearrings.
Finite nearrings without proper subrings are studied in detail under the name of “strictly simple” nearrings. Note that, among the nearrings without proper ideals, we find as “trivial” cases the zero-symmetric nearrings that are just the usual (additive) simple groups equipped with a trivial product, so we do not touch on this study.
A generalization of the idea of strictly simple nearring are the p-singular nearrings, i.e. the nearrings generated by each of its elements of characteristic p (p prime). Such nearrings are studied and classified in detail also using the radical J 2, but a lot of interesting problems on the subject are (often only implicitly) suggested in the main part of this chapter, and we hope to see someday other partial solutions of them.
The other (more or even too wide) generalizations of strictly simple nearrings studied in this chapter are the nearrings with a small number of subnearrings or of ideals and the nearrings generated by a small number of its elements. Once again, contrasting with classical structures, a nearring generated by one element can be very complex.
In the same framework we introduce weakly divisible nearrings because the lattice of their N-subgroups is a chain.
A study on a generalization of integrality (the case in which N contain an integral subset H such that N2 ⫅ H: then N 2 is integral) is introduced, and constructions of some of these nearrings are given.
Celestina Cotti Ferrero, Giovanni Ferrero
Chapter 7. Selfmaps on a Group
Abstract
Today, the studies on the nearrings of maps of a group G to itself are very lively, and usually are very complex because the particular structure of G has a great impact on the structure of such nearrings, and so, only a few of the natural problems arising from this type of topic admits a simple general solution. A classical example is the problem of studying the structure of the nearrings generated by the endomorphisms of G: after older studies (already touched on by [Meldrum, 1985]) there was a long period of time without studies of this type, but now it is possible to find many other paper on the problem; unfortunately such studies are too complex to be used here.
It seems that many nearringer ideas are more suitable to pose new problems of a group-theoretical nature as to some old classical group-theoretical problems.
In the first part of this chapter we introduce and study some generalizations of the endomorphisms of a group: intra-, semi- and quasiendomorphisms, giving some details to show also some examples of elementary techniques for studying nearrings of maps on a group. In particular we like to recall that some studies on nearrings generated by the semi-endomorphisms of a group are linked with a classical well-known problem on the foundations of projective geometry.
In the second part we only touch on some natural topics, mainly giving only some samples of particular results as (very) partial solutions of obvious difficult general problems. Our choosing was mainly guided by the wish to be short and simple, but we also tried to give (when it was possible) some numerative result, because such results may suggest that a satisfactory analysis of certain situations was obtained.
Celestina Cotti Ferrero, Giovanni Ferrero
Chapter 8. Centralizers
Abstract
The studies on the centralizer nearrings are now particularly alive because they have links with other important questions, as we have (only) suggested in the first section on this chapter.
On one extreme, nearrings of homogeneous functions give straightforward generalizations of linearity, and so have an immediate geometric flavor, but they are also linked to the idea of studying the nearrings that are associate (according to definition 1.3.13) to a given nearring. We quote some studies on nearrings of homogeneous functions that are rings and we quote studies on rings that are useful for having a first contact on the studies on associate nearrings. Some examples of results on nonrings of homogeneous functions show just how vast a complete study on such type of questions can be.
On the other extreme there are perhaps the nearrings of the elements commuting with all elements of a group Ф of automorphisms of a group G, these having a role a bit analogous of that of matrix rings in classical theories. These nearrings arose (by Betsch) in the classification of 2- primitive nearrings, but they also have a geometric interest as a generalization of the idea of the multiplier (according to Hall) for designs obtained, for example, by planar nearrings. We try to give some suggestions on the interest of the subject by quoting results (mainly) for cases in which such centralizers are rings and by quoting cases in which Ф is an f.p.f. group. These types of studies can maybe be connected with the interest of many authors (see, for example, [Ke, 1992] and the Clay’s book) on “circular planar nearrings.”
To end we recall some more explicitly geometric ideas: starting by covers or fibrations of a group (see definition 8.4.1) we can construe generalized translation structures, that are geometrical structures where nearrings and centralizers have interesting roles.
Celestina Cotti Ferrero, Giovanni Ferrero
Backmatter
Metadaten
Titel
Nearrings
verfasst von
Celestina Cotti Ferrero
Giovanni Ferrero
Copyright-Jahr
2002
Verlag
Springer US
Electronic ISBN
978-1-4613-0267-4
Print ISBN
978-1-4020-0875-7
DOI
https://doi.org/10.1007/978-1-4613-0267-4