On Quasi-Regular Torsion-Free Modules

A torsion-free module is called quasi-regular if each cyclic submodule is a quasi-summand. This article characterizes torsion-free Abelian groups that are quasi-regular as modules over a subring of their endomorphism ring. In particular, if G is a torsion-free Abelian group such that its ring Q E of quasi-endomorphisms is Artinian, then the left E-module G is quasi-regular if and only if the left C-module G is quasi-regular, where C is the center of its endomorphism ring E.     

Abelian groups as endomorphic modules over their endomorphism ring

Let R be an associative ring with a unit and N be a left R-module. The set M _R(N) = {f: N → N | f(rx) = rf(x), r ∈ R, x ∈ N} is a near-ring with respect to the operations of addition and composition and contains the ring E _R(N) of all endomorphisms of the R-module N. The R-module N is endomorphic if M _R(N) = E _R(N). We call an Abelian group endomorphic if it is an endomorphic module over its endomorphism ring. In this paper, we find endomorphic Abelian groups in the classes of all separable torsion-free groups, torsion groups, almost completely decomposable torsion-free groups, and indecomposable torsion-free groups of rank 2. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 229–233, 2007.     

The quotient field as a torsion-free covering module

R will denote a commutative integral domain with quotient fieldQ. A torsion-free cover of a moduleM is a torsion-free moduleF and anR-epimorphism σ:F→M such that given any torsion-free moduleG and λ∈Hom _R (G, M) there exists μ∈Hom _R (G,F) such that σμ=λ. It is known that ifM is a maximal ideal ofR, R→R/M is a torsion-free cover if and only ifR is a maximal valuation ring. LetE denote the injective hull ofR/M thenR→R/M extends to a homomorphismQ→E. We give necessary and sufficient conditions forQ→E to be a torsion-free cover.     

On Purity and Quasiequality of Abelian Groups

We study the Cohn purity in an abelian group regarded as a left module over its endomorphism ring. We prove that if a finite rank torsion-free abelian group G is quasiequal to a direct sum in which all summands are purely simple modules over their endomorphism rings then the module _E(G) G is purely semisimple. This theorem makes it possible to construct abelian groups of any finite rank which are purely semisimple over their endomorphism rings and it reduces the problem of endopure semisimplicity of abelian groups to the same problem in the class of strongly indecomposable abelian groups.     

Generalizations of coatomic modules

For a ring R and a right R-module M, a submodule N of M is said to be δ-small in M if, whenever N+X=M with M/X singular, we have X=M. Let ℘ be the class of all singular simple modules. Then δ(M)=Σ{ L≤ M| L is a δ-small submodule of M} = Re _jm(℘)=∩{ N⊂ M: M/N∈℘. We call M δ-coatomic module whenever N≤ M and M/N=δ(M/N) then M/N=0. And R is called right (left) δ-coatomic ring if the right (left) R-module R _R(_RR) is δ-coatomic. In this note, we study δ-coatomic modules and ring. We prove M=⊕ _i=1 ⁿ M_i is δ-coatomic if and only if each M _i (i=1,…, n) is δ-coatomic.     

Graded Endomorphism Rings and Equivalences

Abstract

Let R be a G-graded ring,M a G-graded Σ-quasiprojective R- module,and E = END _R (M) its graded ring of endomorphisms. For any subgroup H of G,we prove that certain full subcategories of G/H-graded R-modules associated with M are equivalent to a quotient category of G/H-graded E-modules determined by the idempotent G-graded ideal of E consisting of endomorphisms which factor through a finitely generated submodule of M. Properties and applications of these equivalences are also examined.     

Principally quasi-injective modules

An R-module M is called principally quasi-injective if each R-hornomorphism from a principal submodule of M to M can be extended to an endomorphism of M. Many properties of principally injective rings and quasi-injective modules are extended to these modules. As one application, we show that, for a finite-dimensional quasi-injective module M in which every maximal uniform submodule is fully invariant, there is a bijection between the set of indecomposable summands of M and the maximal left ideals of the endomorphism ring of M

Throughout this paper all rings R are associative with unity, and all modules are unital. We denote the Jacobson radical, the socle and the singular submodule of a module M by J(M), soc(M) and Z(M), respectively, and we write J(M) = J. The notation N ?^ess M means that N is an essential submodule of M.     

𝔏-Rickart Modules

It is well known that the Rickart property of rings is not a left-right symmetric property. We extend the notion of the left Rickart property of rings to a general module theoretic setting and define 𝔏-Rickart modules. We study this notion for a right R-module M _R where R is any ring and obtain its basic properties. While it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an 𝔏-Rickart module is not a left Rickart ring in general. If M _R is a finitely generated 𝔏-Rickart module, we prove that End _R (M) is a left Rickart ring. We prove that an 𝔏-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module. 𝔏-Rickart modules are shown to satisfy a certain kind of nonsingularity which we term “endo-nonsingularity.” Among other results, we prove that M is endo-nonsingular and End _R (M) is a left extending ring iff M is a Baer module and End _R (M) is left cononsingular.     

Irreducible algebraic sets over divisible decomposed rigid groups

A soluble group G is said to be rigid if it contains a normal series of the form G = G ₁ > G ₂ > …> G _p > G _p+1 = 1, whose quotients G _i /G _i+1 are Abelian and are torsion-free when treated as right ℤ[G/G _i ]-modules. Free soluble groups are important examples of rigid groups. A rigid group G is divisible if elements of a quotient G _i /G _i+1 are divisible by nonzero elements of a ring ℤ[G/G _i ], or, in other words, G _i /G _i+1 is a vector space over a division ring Q(G/G _i ) of quotients of that ring. A rigid group G is decomposed if it splits into a semidirect product A ₁ A ₂…A _p of Abelian groups A _i ≅ G _i /G _i+1. A decomposed divisible rigid group is uniquely defined by cardinalities α _i of bases of suitable vector spaces A _i , and we denote it by M(α₁,…, α _p ). The concept of a rigid group appeared in [arXiv:0808.2932v1 [math.GR], ], where the dimension theory is constructed for algebraic geometry over finitely generated rigid groups. In [11], all rigid groups were proved to be equationally Noetherian, and it was stated that any rigid group is embedded in a suitable decomposed divisible rigid group M(α₁,…, α _p ). Our present goal is to derive important information directly about algebraic geometry over M(α₁,… α _p ). Namely, irreducible algebraic sets are characterized in the language of coordinate groups of these sets, and we describe groups that are universally equivalent over M(α₁,…, α _p ) using the language of equations.     

Torsion-free groups and modules with the involution property

An Abelian group or module is said to have the involution property if every endomorphism is the sum of two automorphisms, one of which is an involution. We investigate this property for completely decomposable torsion-free Abelian groups and modules over the ring of p-adic integers.     

Abelian groups with normal endomorphism rings

A ring is said to be normal if all of its idempotents are central. It is proved that a mixed group A with a normal endomorphism ring contains a pure fully invariant subgroup G ⊕ B, the endomorphism ring of a group G is commutative, and a subgroup B is not always distinguished by a direct summand in A. We describe separable, coperiodic, and other groups with normal endomorphism rings. Also we consider Abelian groups in which the square of the Lie bracket of any two endomorphisms is the zero endomorphism. It is proved that every central invariant subgroup of a group is fully invariant iff the endomorphism ring of the group is commutative.     

On modules with <Emphasis Type="Italic">d</Emphasis>-Koszul-type submodules

The so-called weakly d-Koszul-type module is introduced and it turns out that each weakly d-Koszul-type module contains a d-Koszul-type submodule. It is proved that, M ∈ W H J^d（A） if and only if M admits a filtration of submodules： 0 belong to U0 belong to U1 belong to ... belong to Up = M such that all Ui/Ui-1 are d-Koszul-type modules, from which we obtain that the finitistic dimension conjecture holds in W H J^d（A） in a special case. Let M ∈ W H J^d（A）. It is proved that the Koszul dual E（M） is Noetherian, Hopfian, of finite dimension in special cases, and E（M） ∈ gr0（E（A））. In particular, we show that M ∈ W H J^d（A） if and only if E（G（M）） ∈ gr0（E（A））, where G is the associated graded functor.     

On amenable groups of automorphisms on Von Neumann algebras

LetG ⊂ Aut ℳ be a countable group, ℳ a Von Neumann algebra. LetE be a set of pure states on ℳ such thatG*E ⊂E, S ^G be the set ofG invariant states on ℳ andS _E ^G =S ^G ∩w* cl coE. We investigate in this paper some geometric properties for the setS _E ^G which turn out to be equivalent to amenability for the groupG. For example, we show thatS _E ^G ⊂ ℳ_* (S _E ^G has the WRNP) implies that ℳ contains minimal projections (ê containsfinite G invariant orbits) hold true, for all ℳ iffG is amenable. Furthermore we show that ifG is amenable thenS ^G ∩M _* ^⊥ contains a big set, thus improving results obtained by Ching Chou in [2]. These results imply that no action of an amenable countable groupG on an arbitraryW* algebra ℳ iss — strongly ergodic. Moreover cardS ^G ∩M _* ^⊥ ≧2 ^c (see M. Choda [4], K. Schmidt [21] and compare with A. Connes and B. Weiss [5]). The author gratefully acknowledges the support of an Izaak Walton Killam Memorial Senior Fellowship.     

Extending modules relative to a torsion theory

An R-module M is said to be an extending module if every closed submodule of M is a direct summand. In this paper we introduce and investigate the concept of a type 2 τ-extending module, where τ is a hereditary torsion theory on Mod-R. An R-module M is called type 2 τ-extending if every type 2 τ-closed submodule of M is a direct summand of M. If τ _I is the torsion theory on Mod-R corresponding to an idempotent ideal I of R and M is a type 2 τ _I -extending R-module, then the question of whether or not M/MI is an extending R/I-module is investigated. In particular, for the Goldie torsion theory τ _G we give an example of a module that is type 2 τ _G -extending but not extending.     

A combinatorial principle and endomorphism rings I: Onp-groups

Two lines of research are involved here. One is a combinatorial principle, proved in ZFC for many cardinals (e.g., any λ = λ^ℵ ₀) enabling us to prove things which have been proven using the diamond or for strong limit cardinals of uncountable cofinality. The other direction is building abelian groups with few endomorphisms and/or prescribed rings of endomorphisms. We prove that for a ringR, whose additive group is thep-adic completion of a freep-adic module,R is isomorphic to the endomorphism ring of some separable abelianp-groupG divided by the ideal of small endomorphisms, withG of power λ for any λ = λ^ℵ ₀≧|R|. Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death The author would like to thank the United States-Israel Binational Science Foundation for partially supporting this research.     

Relative purity over noetherian rings

In this note we are going to show that if M is a left module over a left noetherian ring R of the infinite cardinality λ ≥ |R|, then its injective hull E(M) is of the same size. Further, if M is an injective module with |M| ≥ (2^λ)⁺ and K ≤ M is its submodule such that |M/K| ≤ λ, then K contains an injective submodule L with |M/L| ≤ 2^λ. These results are applied to modules which are torsionfree with respect to a given hereditary torsion theory and generalize the results obtained by different methods in author’s previous papers: [A note on pure subgroups, Contributions to General Algebra 12. Proceedings of the Vienna Conference, June 3–6, 1999, Verlag Johannes Heyn, Klagenfurt, 2000, pp. 105–107], [Pure subgroups, Math. Bohem. 126 (2001), 649–652]. This research has been partially supported by the Grant Agency of the Charles University, grant #GAUK 301-10/203115/B-MAT/MFF and also by the institutional grant MSM 113 200 007.     

Endomorphism algebras of transitive permutation modules for p-groups

Let G be a finite p-group with subgroup H and k a field of characteristic p. We study the endomorphism algebra E = End_kG(k_H ↑^G), showing that it is a split extension of a nilpotent ideal by the group algebra kN_G(H)/H. We identify the space of endomorphisms that factor through a projective kG-module and hence the endomorphism ring of k_H ↑^G in the stable module category, and determine the Loewy structure of E when G has nilpotency class 2 and [G, H] is cyclic. Received: 3 November 2008     

Torsion-free modules with UA-rings of endomorphisms

An associative ring R is called a unique addition ring (UA-ring) if its multiplicative semigroup (R, · ) can be equipped with a unique binary operation+ transforming the triple (R, ·, +) to a ring. An R-module A is said to be an End-UA-module if the endomorphism ring End _R (A) of A is a UA-ring. In the paper, the torsion-free End-UA-modules over commutative Dedekind domains are studied. In some classes of Abelian torsion-free groups, the Abelian groups having UA-endomorphism rings are found.     

Universal theories for rigid soluble groups

A group is said to be p-rigid, where p is a natural number, if it has a normal series of the form G = G ₁ > G ₂ > … > G _p  > G _p+1 = 1, whose quotients G _i /G _i+1 are Abelian and are torsion free when treated as \mathbbZ \mathbb{Z} [G/G _i ]-modules. Examples of rigid groups are free soluble groups. We point out a recursive system of universal axioms distinguishing p-rigid groups in the class of p-soluble groups. It is proved that if F is a free p-soluble group, G is an arbitrary p-rigid group, and W is an iterated wreath product of p infinite cyclic groups, then ∀-theories for these groups satisfy the inclusions A(F) ê A(G) ê A(W) \mathcal{A}(F) \supseteq \mathcal{A}(G) \supseteq \mathcal{A}(W) . We construct an ∃-axiom distinguishing among p-rigid groups those that are universally equivalent to W. An arbitrary p-rigid group embeds in a divisible decomposed p-rigid group M = M(α₁,…, α _p ). The latter group factors into a semidirect product of Abelian groups A ₁ A ₂…A _p , in which case every quotient M _i /M _i+1 of its rigid series is isomorphic to A _i and is a divisible module of rank α_i over a ring \mathbbZ \mathbb{Z} [M/M _i ]. We specify a recursive system of axioms distinguishing among M-groups those that are Muniversally equivalent to M. As a consequence, it is stated that the universal theory of M with constants in M is decidable. By contrast, the universal theory of W with constants is undecidable.     

On weakly prime radical of modules and semi-compatible modules

Summary Let M be a left R-module. Then a proper submodule P of M is called weakly prime submodule if for any ideals A and B of R and any submodule N of M such that ABN ⊆ P, we have AN ⊆ P or BN ⊆ P. We define weakly prime radicals of modules and show that for Ore domains, the study of weakly prime radicals of general modules reduces to that of torsion modules. We determine the weakly prime radical of any module over a commutative domain R with dim (R) ≦ 1. Also, we show that over a commutative domain R with dim (R) ≦ 1, every semiprime submodule of any module is an intersection of weakly prime submodules. Localization of a module over a commutative ring preserves the weakly prime property. An R-module M is called semi-compatible if every weakly prime submodule of M is an intersection of prime submodules. Also, a ring R is called semi-compatible if every R-module is semi-compatible. It is shown that any projective module over a commutative ring is semi-compatible and that a commutative Noetherian ring R is semi-compatible if and only if for every prime ideal B of R, the ring R/\B is a Dedekind domain. Finally, we show that if R is a UFD such that the free R-module R⊕ R is a semi-compatible module, then R is a Bezout domain.