Pure-injective Modules Research Articles

Relative injective modules, superstability and noetherian categories

We study classes of modules closed under direct sums, [Formula: see text]-submodules and [Formula: see text]-epimorphic images where [Formula: see text] is either the class of embeddings, RD-embeddings or pure embeddings. We show that the [Formula: see text]-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, RD-injective modules, pure injective modules, flat cotorsion modules and [Formula: see text]-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah’s non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability.

Wide coreflective subcategories and torsion pairs

We revisit a construction of wide subcategories going back to work of Ingalls and Thomas. To a torsion pair in the category R−mod of finitely presented modules over a left artinian ring R, we assign two wide subcategories in the category R−Mod of all R-modules and describe them explicitly in terms of an associated cosilting module. It turns out that these subcategories are coreflective, and we address the question of which wide coreflective subcategories can be obtained in this way. Over a tame hereditary algebra, they are precisely the categories which are perpendicular to collections of pure-injective modules.

Pure projective, pure injective and FP-injective modules over trivial ring extensions

Let [Formula: see text] be a trivial extension of a ring [Formula: see text] by an [Formula: see text]-[Formula: see text]-bimodule [Formula: see text]. We first study the properties of pure projective and pure injective modules over [Formula: see text]. Then we characterize [Formula: see text]-injective modules over [Formula: see text]. Finally some applications are given to Morita context rings.

Classification of 1-absorbing comultiplication modules over a pullback ring

One of the aims of the modern representation theory is to solve classification problems for subcategories of modules over a unitary ring R. In this paper, we introduce the concept of 1-absorbing comultiplication modules and classify 1-absorbing comultiplication modules over local Dedekind domains and we study it in detail from the classification problem point of view. The main purpose of this article is to classify all those indecomposable 1-absorbing comultiplication modules with finite-dimensional top over pullback rings of two local Dedekind domains and establish a connection between the 1-absorbing comultiplication modules and the pure-injective modules over such rings. In fact, we extend the definition and results given in [17] to a more general 1-absorbing comultiplication modules case.

Characterizing categoricity in several classes of modules

We show that the condition of being categorical in a tail of cardinals can be characterized algebraically for several classes of modules. Theorem 0.1Assume R is an associative ring with unity.(1)The class of locally pure-injective R-modules is λ-categorical in all λ>card(R)+ℵ0if and only ifR≅Mn(D)for D a division ring andn≥1.(2)The class of flat R-modules is λ-categorical in all λ>card(R)+ℵ0if and only ifR≅Mn(k)for k a local ring such that its maximal ideal is left T-nilpotent andn≥1.(3)Assume R is a commutative ring. The class of absolutely pure R-modules is λ-categorical in all λ>card(R)+ℵ0if and only if R is a local artinian ring.We show that in the above results it is enough to assume λ-categoricity in some big cardinal λ. This shows that Shelah's Categoricity Conjecture holds for the class of locally pure-injective modules, flat modules and absolutely pure modules. These classes are not first-order axiomatizable for arbitrary rings.We provide rings such that the class of flat modules is categorical in a tail of cardinals but it is not first-order axiomatizable.

Maranda’s theorem for pure-injective modules and duality

AbstractLet R be a discrete valuation domain with field of fractions Q and maximal ideal generated by $\pi $ . Let $\Lambda $ be an R-order such that $Q\Lambda $ is a separable Q-algebra. Maranda showed that there exists $k\in \mathbb {N}$ such that for all $\Lambda $ -lattices L and M, if $L/L\pi ^k\simeq M/M\pi ^k$ , then $L\simeq M$ . Moreover, if R is complete and L is an indecomposable $\Lambda $ -lattice, then $L/L\pi ^k$ is also indecomposable. We extend Maranda’s theorem to the class of R-reduced R-torsion-free pure-injective $\Lambda $ -modules.As an application of this extension, we show that if $\Lambda $ is an order over a Dedekind domain R with field of fractions Q such that $Q\Lambda $ is separable, then the lattice of open subsets of the R-torsion-free part of the right Ziegler spectrum of $\Lambda $ is isomorphic to the lattice of open subsets of the R-torsion-free part of the left Ziegler spectrum of $\Lambda $ .Furthermore, with k as in Maranda’s theorem, we show that if M is R-torsion-free and $H(M)$ is the pure-injective hull of M, then $H(M)/H(M)\pi ^k$ is the pure-injective hull of $M/M\pi ^k$ . We use this result to give a characterization of R-torsion-free pure-injective $\Lambda $ -modules and describe the pure-injective hulls of certain R-torsion-free $\Lambda $ -modules.

Quasi semiprime multiplication modules over a pullback of a pair of Dedekind domains

The main purpose of this article is to classify all indecomposable quasi semiprime multiplication modules over pullback rings of two Dedekind domains and establish a connection between the quasi semiprime multiplication modules and the pure-injective modules over such rings. First, we introduce and study the notion of quasi semiprime multiplication modules and classify quasi semiprime multiplication modules over local Dedekind domains. Second, we get all indecomposable separated quasi semiprime multiplication modules and then, using this list of separated quasi-semiprime multiplication modules, we show that non-separated indecomposable quasi semiprime multiplication R-modules with finite-dimensional top are factor modules of finite direct sums of separated indecomposable quasi semiprime multiplication modules.

On subinjectivity domains of pure-injective modules

A pure-injective module M is said to be pi-indigent if its subinjectivity domain is smallest possible, namely, consisting of exactly the absolutely pure modules. A module M is called subinjective relative to a module N if for every extension K of N, every homomorphism N→M can be extended to a homomorphism K→M. The subinjectivity domain of the module M is defined to be the class of modules N such that M is N-subinjective. Basic properties of the subinjectivity domains of pure-injective modules and of pi-indigent modules are studied. The structure of a ring over which every simple, uniform, or indecomposable pure-injective module is injective or subinjective relative only to the smallest possible family of modules is investigated.

On the isomorphism of coneat-injective modules over commutative rings

It is well-known that the family of coneat-injective modules has many properties that resemble the classes of injective modules, pure-injective modules and RD-injective modules. Moreover, injective modules are coneat-injective which, in turn, are RD-injective and, finally, RD-injective modules are pure-injective. For injective (respectively, pure-injective, RD-injective) modules, two such modules which are isomorphic to submodules (respectively, pure submodules, relatively divisible submodules) of each other are isomorphic. Motivated by these results, this communication is devoted to demonstrating that two coneat-injective modules which are isomorphic to coneat submodules of each other are likewise isomorphic.

KARAKTERISASI MODUL CYCLICALLY PURE (CP)-INJEKTIF

This research deals with the structure of cyclically pure injective modules over a commutative ring R. If I be an ideal of R, proved that any CP-injective R/Imodul is also CP-injective as an R-module. The main result of research is the existance of CP-injective R-module if there is an R-module. More over, we deal characterization of CP-injective module which is related to proper essential ctclically pure extension. It is shown that R-modul D is cyclically pure injective if and only if D has no proper essential cyclically pure extension.

On Krull-Gabriel dimension and Galois coverings

Assume that K is an algebraically closed field, R a locally support-finite locally bounded K-category, G a torsion-free admissible group of K-linear automorphisms of R and A=R/G. We show that the Krull-Gabriel dimension KG(R) of R equals the Krull-Gabriel dimension KG(A) of A. Furthermore, we determine the Krull-Gabriel dimension of locally support-finite repetitive K-categories. These results are applied to determine the Krull-Gabriel dimension of standard selfinjective algebras of polynomial growth. Finally, we show that there are no super-decomposable pure-injective modules over standard selfinjective algebras of domestic type.

ABSORBING COMULTIPLICATION MODULES OVER A PULLBACK RING

The purpose of this paper is to introduce the concept of 2-absorbing comultiplication modules, which form a subclass of the class of pure-injective modules over pullback rings. A full description of all indecomposable 2- absorbing comultiplication modules with nite-dimensional top over the pull- back of two discrete valuation domains with the same residue eld is given.

Pseudo-absorbing multiplication modules over a pullback ring

In this paper, we introduce the notion of pseudo-absorbing multiplication modules. The purpose of this paper is to outline an approach to the classification of indecomposable pseudo-absorbing multiplication modules with finite-dimensional top over certain kinds of pullback rings and establish a connection between the pseudo-absorbing multiplication modules and the pure-injective modules over such rings.

Weak-injective modules over commutative rings

ABSTRACTWeak-injective modules were defined as modules M for which holds for all modules N of weak dimension ≤1. They were studied over integral domains in several papers. Here we consider them over arbitrary commutative rings, especially their relations to h-divisible, pure-injective, absolutely pure, and flat modules. It turns out that they retain some of the features of the domain case even if the underlying rings admit zero-divisors, but several useful properties are not available any more.

A note on the stability of pure-injective modules

ABSTRACTLet R be a ring. Then a left R-module N is pure-injective if and only if HomR(M,N) is a pure-injective left S-module for any ring S and any (R,S)-bimodule . If R is a commutative ring and M,N are R-modules with N pure-injective, then is a pure-injective R-module for any n≥0. Let R and S be rings and let be an (S,R)-bimodule and M a finitely presented left R-module. If N is pure-injective as a left S-module, then the left S-module N⊗RM is pure-injective; and if R is left coherent, then the left S-module is pure-injective for any n≥1.

An equivalence criterion for the generalized injectivity of modules with respect to algebraic classes of homomorphisms

Departing from a general definition of injectivity of modules with respect to suitable algebraic classes of morphisms, we establish conditions under which two modules are isomorphic when they are isomorphic to submodules of each other. The main result of this work extends both Bumby’s criterion for the isomorphism of injective modules and the well-known Cantor–Bernstein–Schröder’s theorem on the cardinality of sets. In the way, various properties on essential extensions, injective modules and injective hulls are generalized. The applicability of our main theorem embraces the cases of [Formula: see text]-injective and pure-injective modules as particular scenarios. Many of the propositions which lead to the proof of the main result of this paper are valid for arbitrary categories.

Generalized Injectivity and Approximations

Injective, pure-injective, and fp-injective modules are well known to provide for approximations in the category Mod-R for an arbitrary ring R. We prove that this fails for many other generalizations of injectivity: the C1, C2, C3, quasi-continuous, continuous, and quasi-injective modules. We show that, except for the class of all C1-modules, each of the latter classes provides for approximations only when it coincides with the injectives (for quasi-injective modules, this forces R to be a right noetherian V-ring; in the other cases, R even has to be semisimple artinian). The class of all C1-modules over a right noetherian ring R is (pre)enveloping iff R is a certain right artinian ring of Loewy length ≤2; in this case, however, R may have an arbitrary representation type.

Powers of the phantom ideal

It is proved that if G is a finite group, then the order of G is a proper upper bound for the phantom number of G. More specifically, if k is a field whose characteristic divides the order of G, and $\Phi$ is the ideal of phantom morphisms in the stable category k[G]-$\underline{\rm Mod}$ of modules over the group algebra k[G], then $\Phi^{n-1} = 0,$ where n is the nilpotency index of the Jacobson radical J of k[G]. If $R$ is a semiprimary ring, with $J^n =0,$ and $\Phi$ denotes the phantom ideal in the module category R-Mod, then $\Phi^n$ is the ideal of morphisms that factor through a projective module. If R is a right coherent ring and every cotorsion left R-module has a coresolution of length $n$ by pure injective modules, then $\Phi^{n+1}$ is the ideal of morphisms that factor through a flat module. These results are obtained by introducing the mono-epi (ME) exact structure on the morphisms of an exact category (${\mathcal A}; {\mathcal E}$), used to prove new versions of Salce's Lemma, the Ghost Lemma of Christensen, and Wakamatsu's Lemma. Salce's Lemma gives a bijective correspondence between special precovering ideals and special preenveloping ideals. The exact category (Arr (${\mathcal A}$); ME) of morphisms allows us to introduce the notion of an extension $i \star j$ of morphisms and the notion of an extension of ideals in (${\mathcal A}; {\mathcal E}$). The Ghost Lemma asserts that the class of special precovering (resp., special preenveloping) ideals is closed under products and extensions and that the bijective correspondence of Salce's Lemma replaces multiplication with extension. Wakamatsu's Lemma is the statement that if a covering ideal is closed under extensions of morphisms, then it is a special precovering ideal with a syzygy ideal generated by objects.

Super-decomposable pure-injective modules over algebras with strongly simply connected Galois coverings

Assume that k is a field of characteristic different from 2, R is a strongly simply connected locally bounded k-category, G is an admissible group of k-linear automorphisms of R and A=R/G. We show that if A is not of polynomial growth, then there exists a special family of pointed A-modules, called an independent pair of dense chains of pointed modules. Then it follows by a result of M. Ziegler that there exists a super-decomposable pure-injective A-module, if the field k is countable. We also prove that if A is of non-polynomial growth and R does not contain a convex subcategory which is hypercritical or pg-critical, then there exists a factor algebra B of A such that B is a string algebra of non-polynomial growth. This fact is one of the key elements in the proof of our main results.

Konstruksi Ring Bersih dari Sebarang Ring

The aims of this research was to construct a clean ring from any ring. The base on the fact that the endomorphism ring of every pure-injective module is clean, it was constructed a clean ring from any ring. So, the result of this research was it always could be constructed a clean ring from any ring.