The low-dimensional homology of projective linear group of degree two

Let GL2(R) be the general linear group of 2 × 2 invertible matrices in M2(R) over a commutative ring R with 1 and SL2(R) be the special linear group consisting of 2 × 2 matrices over R with determinant 1. In this paper we determine the homomorphisms from GL2 and SL2, as well as their projective groups, over Laurent polynomial rings to those groups over Gaussian domains, i.e. unique factorization domains (cf. Theorems 1, 2, 3 below). We also consider more generally the homomorphisms of non-projective groups over commutative rings containing a field which are generated by their units (cf. Theorems 4 and 5). So far the homomorphisms of two-dimensional linear groups over commutative rings have only been studied in some specific cases. Landin and Reiner[7] obtained the automorphisms of GL2(R), where R is a Euclidean domain generated by its units. When R is a type of generalized Euclidean domain with a degree function and with units of R and 0 forming a field, Cohn[3] described the automorphisms of GL2(R). Later, Cohn[4] applied his methods to the case of certain rings of quadratic integers. Dull[6] has considered the automorphisms of GL2(R) and SL2(R), along with their projective groups, provided that R is a GE-ring and 2 is a unit in R. McDonald [9] examined the automorphisms of GL2(R) when R has a large unit group. The most recent work of which we are aware is that of Li and Ren[8] where the automorphisms of E2(R) and GE2(R) were determined for any commutative ring R in which 2, 3 and 5 are units.

This book is an introduction to the theory of associative rings and their modules, designed primarily for graduate students. The standard topics on the structure of rings are covered, with a particular emphasis on the concept of the complete ring of quotients. A survey of the fundamental concepts of algebras in the first chapter helps to make the treatment self-contained. The topics covered include selected results on Boolean and other commutative rings, the classical structure theory of associative rings, injective modules, and rings of quotients. The final chapter provides an introduction to homological algebra. Besides three appendices on further results, there is a six-page section of historical comments. Table of Contents: Fundamental Concepts of Algebra: 1.1 Rings and related algebraic systems; 1.2 Subrings, homomorphisms, ideals; 1.3 Modules, direct products, and direct sums; 1.4 Classical isomorphism theorems. Selected Topics on Commutative Rings: 2.1 Prime ideals in commutative rings; 2.2 Prime ideals in special commutative rings; 2.3 The complete ring of quotients of a commutative ring; 2.4 Rings of quotients of commutative semiprime rings; 2.5 Prime ideal spaces. Classical Theory of Associative Rings: 3.1 Primitive rings; 3.2 Radicals; 3.3 Completely reducible modules; 3.4 Completely reducible rings; 3.5 Artinian and Noetherian rings; 3.6 On lifting idempotents; 3.7 Local and semiperfect rings. Injectivity and Related Concepts: 4.1 Projective modules; 4.2 Injective modules; 4.3 The complete ring of quotients; 4.4 Rings of endomorphisms of injective modules; 4.5 Regular rings of quotients; 4.6 Classical rings of quotients; 4.7 The Faith-Utumi theorem. Introduction to Homological Algebra: 5.1 Tensor products of modules; 5.2 Hom and $\otimes$ as functors; 5.3 Exact sequences; 5.4 Flat modules; 5.5 Torsion and extension products. Appendixes; Comments; Bibliography; Index. Review from Zentralblatt Math: Due to their clarity and intelligible presentation, these lectures on rings and modules are a particularly successful introduction to the surrounding circle of ideas. Review from American Mathematical Monthly: An introduction to associative rings and modules which requires of the reader only the mathematical maturity which one would attain in a first-year graduate algebra [course]...in order to make the contents of the book as accessible as possible, the author develops all the fundamentals he will need. In addition to covering the basic topics...the author covers some topics not so readily available to the nonspecialist...the chapters are written to be as independent as possible...[which will be appreciated by] students making their first acquaintance with the subject...one of the most successful features of the book is that it can be read by graduate students with little or no help from a specialist. (CHEL/283.H)

Unique decomposition into ideals for commutative Noetherian rings

This paper has as its aim the geometric description of three important groups classically associated with a commutative ring R as the first (and only) three non-trivial homotopy groups of a reduced (and hence connected) simplicial Kan-complex AZ(R) which we wi l l call the Azumaya complex of the ring. This description, which places the Brauer group Br(R) of equivalence classes of Azumaya algebras over the ring as T~, of the complex, the Picard group Pic(R) of isomorphism classes of invertible modules over it as rr a, and the multiplicative group Gm(R) [= R x, the group of units] of the ring as W3, is stable under arbitrary change of the commutative base ring and (among other things) allows the recovery of several wel l-known exact sequences linking these groups as instances of exact sequences of groups obtained in simplicial homotopy theory. This conference paper wi l l be devoted to a direct description of the simplicial complex itself wi th the applications which involve the computation of a particular spectral sequence to appear elsewhere. The complex we construct here is, in fact, one which is associated with a particular 3-category AZI~ which has a single object (i.e.,0-ceil) the commutative ring R. Its 1-cells are Azumaya algebras over R, A R )R 5

Cycle indices of linear, affine, and projective groups

In a recent paper, Mao has studied min-pure injective modules to investigate the existence of min-injective covers. A min-pure injective module is one that is injective relative only to min-pure exact sequences. In this paper, we study the notion of min-pure projective modules which is the projective objects of min-pure exact sequences. Various ring characterizations and examples of both classes of modules are obtained. Along this way, we give conditions which guarantee that each min-pure projective module is either injective or projective. Also, the rings whose injective objects are min-pure projective are considered. The commutative rings over which all injective modules are min-pure projective are exactly quasi-Frobenius. Finally, we are interested with the rings all of its modules are min-pure projective. We obtain that a ring $R$ is two-sided K\"othe if all right $R$-modules are min-pure projective. Also, a commutative ring over which all modules are min-pure projective is quasi-Frobenius serial. As consequence, over a commutative indecomposable ring with $J(R)^{2}=0$, it is proven that all $R$-modules are min-pure projective if and only if $R$ is either a field or a quasi-Frobenius ring of composition length $2$.

Let R be a commutative ring with identity. A short exact sequence of R-modules is said to be neat (respectively, -pure) if the sequence (respectively, ) is exact for every simple R-module S. The characterization of the integral domains over which neatness and -purity coincide has been given by László Fuchs: they are the integral domains where every maximal ideal is projective (and so also finitely generated). We show that for a commutative ring R, every maximal ideal of R is finitely generated and projective if and only if R has projective socle and neatness and -purity coincide. For a commutative ring R, we prove that neatness and -purity coincide if and only if every maximal ideal of R is finitely generated and the unique maximal ideal PP of the local ring RP is a principal ideal for every maximal ideal P of R. This result is proved firstly over commutative local rings and then using localization over any commutative ring. The Auslander-Bridger transpose of simple modules is used in proving these equivalences because it is an essential tool for the passage between proper classes of short exact sequences of modules that are projectively generated and these that are flatly generated by a set of finitely presented modules. For a commutative ring R, we prove that neatness and -purity coincide if and only if every simple R-module S is finitely presented and an Auslander-Bridger transpose of S is projectively equivalent to S.

A module over a ring R is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module M, we consider Add ⁡ ( M ) {\operatorname{Add}(M)} , which consists of direct summands of direct sums of copies of M. We are primarily interested in the case where R is a one-dimensional, local domain, and in torsion-free (or Cohen–Macaulay) modules. We show that, even in this case, Add ⁡ ( M ) {\operatorname{Add}(M)} can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of M and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid V * ⁢ ( M ) {V^{*}(M)} , of isomorphism classes of countably generated modules in Add ⁡ ( M ) {\operatorname{Add}(M)} with the addition induced by the direct sum. We show that V * ⁢ ( M ) {V^{*}(M)} is a submonoid of V * ⁢ ( M ⊗ R R ^ ) {V^{*}(M\otimes_{R}\widehat{R})} , this allows us to make computations with examples and to prove some realization results.

Complex linear groups of degree less than 2 [formula omitted] p — 3

In this article we study the low dimensional homology groups of the special linear group SL 2 ( A ) and the projective special linear group PSL 2 ( A ) , A a domain, through the natural surjective map SL 2 ( A ) → PSL 2 ( A ) . In particular, we study the connection of the first, the second, and the third homology groups of these groups over euclidean domains Z [ 1 / m ] , m a square free integer, and local domains.

In this paper, we consider the projective special linear group $PSL_2(59)$ and construct some 1-designs by applying the Key-Moori method on $PSL_2(59)$. Moreover, we obtain parameters of these designs and their automorphism groups. It is shown that $PSL_2(59)$ and $PSL_2(59):2$ appear as the automorphism group of the constructed designs.

The results of the paper relate to the following general problem. Find natural finite generating sets of elements of a given linear group over a finitely generated commutative ring. Of particular interest are coefficient rings that are generated by a single element, for example, the ring of integers or the ring of Gaussian integers. We prove that a projective general linear group of dimension n over the ring of Gaussian integers is generated by three involutions two of which commute if and only if n is greater than 4 and 4 does not divide n. Earlier, M. A. Vsemirnov, R. I. Gvozdev, D. V. Levchuk and the authors of this paper solved a similar problem for the special and projective special linear groups.

Suppose R is a commutative ring with 1, б=(б ij ) is a fixedD-net of ideals of R of ordern, and Gб is the corresponding net subgroup of the general linear group GL (n, R). There is constructed for б a homomorphismdet б of the subgroup G(б) into a certain Abelian group Φ(б). Let I be the index set {1...,n}. For each subset α⫅I let б(∝)=∑б ij б ji , wherei, ranges over all indices in α and j independently over the indices in the complement Iα (б(I) is the zero ideal). Letdet ∝(a) denote the principal minor of order |α|⩽n of the matrixa ∃ G (б) corresponding to the indices in α, and let' Φ(б) be the Cartesian product of the multiplicative groups of the quotient rings R/б(α) over all subsets α⫅ I. The homomorphismdet б is defined as follows: It is proved that if R is a semilocal commutative Bezout ring, then the kernelKer det б coincides with the subgroup E(б) generated by all transvections in G(б). For these R is also definedTm det б.

We prove that a ring [Formula: see text] has a module [Formula: see text] whose domain of projectivity consists of only some injective modules if and only if [Formula: see text] is a right noetherian right [Formula: see text]-ring. Also, we consider modules which are projective relative only to a subclass of max modules. Such modules are called max-poor modules. In a recent paper Holston et al. showed that every ring has a p-poor module (that is a module whose projectivity domain consists precisely of the semisimple modules). So every ring has a max-poor module. The structure of all max-poor abelian groups is completely determined. Examples of rings having a max-poor module which is neither projective nor p-poor are provided. We prove that the class of max-poor [Formula: see text]-modules is closed under direct summands if and only if [Formula: see text] is a right Bass ring. A ring [Formula: see text] is said to have no right max-p-middle class if every right [Formula: see text]-module is either projective or max-poor. It is shown that if a commutative noetherian ring [Formula: see text] has no right max-p-middle class, then [Formula: see text] is the ring direct sum of a semisimple ring [Formula: see text] and a ring [Formula: see text] which is either zero or an artinian ring or a one-dimensional local noetherian integral domain such that the quotient field [Formula: see text] of [Formula: see text] has a proper [Formula: see text]-submodule which is not complete in its [Formula: see text]-topology. Then we show that a commutative noetherian hereditary ring [Formula: see text] has no right max-p-middle class if and only if [Formula: see text] is a semisimple ring.

We introduce ⊕ -radical supplemented modules and strongly ⊕ -radical supplemented modules (briefly, srs ⊕-modules) as proper generalizations of ⊕ -supplemented modules. We prove that (1) a semilocal ring R is left perfect if and only if every left R-module is an ⊕ -radical supplemented module; (2) a commutative ring R is an Artinian principal ideal ring if and only if every left R-module is an srs ⊕-module; (3) over a local Dedekind domain, every ⊕ -radical supplemented module is an srs ⊕-module. Moreover, we completely determine the structure of these modules over local Dedekind domains.