Pure Injective Envelopes Research Articles

Absolutely pure covers of sheaves

Let [Formula: see text] be a scheme and [Formula: see text] be the category of all sheaves of [Formula: see text]-modules. Enochs’ conjecture states that any covering class in [Formula: see text] is closed under directed colimits. In this work, if Abs[Formula: see text] is the class of all absolutely pure objects in [Formula: see text], then we prove the validity of the conjecture for Abs[Formula: see text]. This gives a characterization of locally coherent schemes. In addition, we study some homological properties of absolutely pure sheaves of [Formula: see text]-modules and answer a question raised in [P. Rothmaler, When are pure injective envelopes of flat modules flat? Comm. Algebra 30(6) (2002) 3077–3085].

SOME STABLE NON-ELEMENTARY CLASSES OF MODULES

AbstractFisher [10] and Baur [6] showed independently in the seventies that if T is a complete first-order theory extending the theory of modules, then the class of models of T with pure embeddings is stable. In [25, 2.12], it is asked if the same is true for any abstract elementary class $(K, \leq _p)$ such that K is a class of modules and $\leq _p$ is the pure submodule relation. In this paper we give some instances where this is true:Theorem.Assume R is an associative ring with unity. Let $(K, \leq _p)$ be an AEC such that $K \subseteq R\text {-Mod}$ and K is closed under finite direct sums, then: •If K is closed under pure-injective envelopes, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda \geq \operatorname {LS}(\mathbf {K})$ such that $\lambda ^{|R| + \aleph _0}= \lambda $ .•If K is closed under pure submodules and pure epimorphic images, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda $ such that $\lambda ^{|R| + \aleph _0}= \lambda $ .•Assume R is Von Neumann regular. If $\mathbf {K}$ is closed under submodules and has arbitrarily large models, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda $ such that $\lambda ^{|R| + \aleph _0}= \lambda $ .As an application of these results we give new characterizations of noetherian rings, pure-semisimple rings, Dedekind domains, and fields via superstability. Moreover, we show how these results can be used to show a link between being good in the stability hierarchy and being good in the axiomatizability hierarchy.Another application is the existence of universal models with respect to pure embeddings in several classes of modules. Among them, the class of flat modules and the class of $\mathfrak {s}$ -torsion modules.

Modules Coinvariant Under the Idempotent Endomorphisms of Their Covers

We study the modules that are coinvariant under the idempotent endomorphisms of their covers. Some generalizations of discrete and continuous modules are introduced and inspected on using the theory of covers and envelopes of modules. By way of application, we consider the cases of flat covers, injective envelopes, and pure injective envelopes.

The Schröder–Bernstein problem for modules

In this paper we study the Schröder–Bernstein problem for modules. We obtain a positive solution for the Schröder–Bernstein problem for modules invariant under endomorphisms of their general envelopes under some mild conditions that are always satisfied, for example, in the case of injective, pure-injective or cotorsion envelopes. In the particular cases of injective envelopes and pure-injective envelopes, we are able to extend it further and we show that the Schröder–Bernstein problem has a positive solution even for modules that are invariant only under automorphisms of their injective envelopes or pure-injective envelopes.

Pure Injective and Absolutely Pure Sheaves

AbstractWe study two notions of purity in categories of sheaves: the categorical and the geometric. It is shown that pure injective envelopes exist in both cases under very general assumptions on the scheme. Finally, we introduce the class of locally absolutely pure (quasi-coherent) sheaves with respect to the geometrical purity, and characterize locally Noetherian closed subschemes of a projective scheme in terms of the new class.

Modules invariant under automorphisms of their covers and envelopes

In this paper we develop a general theory of modules which are invariant under automorphisms of their covers and envelopes. When applied to specific cases like injective envelopes, pure-injective envelopes, cotorsion envelopes, projective covers, or flat covers, these results extend and provide a much more succinct and clear proofs for various results existing in the literature. Our results are based on several key observations on the additive unit structure of von Neumann regular rings.

The existence of relative pure injective envelopes

Let $\mathcal{S}$ be a class of finitely presented $R$-modules such that $R\in \mathcal{S}$ and $\mathcal{S}$ has a subset $\mathcal{S}^*,$ with the property that for any $U\in \mathcal{S}$ there is a $U^*\in \mathcal{S}^*$ with $U^*\cong U.$ We show that the class of $\mathcal{S}$-pure injective $R$-modules is preenveloping. As an application, we deduce that the left global $\mathcal{S}$-pure projective dimension of $R$ is equal to its left global $\mathcal{S}$-pure injective dimension. As our main result, we prove that, in fact, the class of $\mathcal{S}$-pure injective $R$-modules is enveloping.

Characterizations of Chain Conditions on Finite Matrix Subgroups

We give several characterizations of Zimmermann's uniform chain condition on matrix subgroups. In particular, we prove that a family of modules satisfies it precisely when the direct sum of their pure injective envelopes splits in their direct product. We also prove that a weaker non-uniform version of this condition is equivalent to Σ-algebraic compactness of reduced products.

Modules with Cosupport and Injective Functors

Several authors have studied the filtered colimit closure \(\varinjlim\mathcal{B}\) of a class \(\mathcal{B}\) of finitely presented modules. Lenzing called \(\varinjlim\mathcal{B}\) the category of modules with support in \(\mathcal{B}\), and proved that it is equivalent to the category of flat objects in the functor category \((\mathcal{B}^\mathrm{op},\mathsf{Ab})\). In this paper, we study the category \(({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}\) of modules with cosupport in \(\mathcal{B}\). We show that \(({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}\) is equivalent to the category of injective objects in \((\mathcal{B},\mathsf{Ab})\), and thus recover a classical result by Jensen-Lenzing on pure injective modules. Works of Angeleri-Hügel, Enochs, Krause, Rada, and Saorín make it easy to discuss covering and enveloping properties of \(({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}\), and furthermore we compare the naturally associated notions of \(\mathcal{B}\)-coherence and \(\mathcal{B}\)-noetherianness. Finally, we prove a number of stability results for \(\varinjlim\mathcal{B}\) and \(({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}\). Our applications include a generalization of a result by Gruson-Jensen and Enochs on pure injective envelopes of flat modules.

SOME REMARKS ON COTORSION ENVELOPES OF MODULES

In this paper we prove that the extension of pure injective module is pure injective if and only if the cotorsion envelope and the pure injective envelope of any R-module M are isomorphic over M . And we prove that if the product of pure injective envelopes of flat modules is a pure injective envelope and the product of flat covers is a flat cover, then the product of cotorsion envelopes is a cotorsion envelope.

How to construct a ‘concrete’ superdecomposable pure-injective module over a string algebra

We construct an element in a direct product of finite dimensional modules over a string algebra such that the pure-injective envelope of this element is a superdecomposable module.

Applications of Duality to the Pure-injective Envelope

Given an R-T-bimodule R K T and R-S-bimodule R M S , we study how properties of R K T affect the K-double dual M** = Hom T [Hom R (M, K), K] considered as a right S-module. If R K is a cogenerator, then for every R-S-bimodule, the natural morphism Φ M : M → M** is a pure-monomorphism of right S-modules. If R K is the minimal (injective) cogenerator and K T is quasi-injective, then M ** is a pure-injective right S-module. If R K is the minimal (injective) cogenerator, and T = End R K it is shown that K T is quasi-injective if and only if the K-topology on R is linearly compact. If the R K-topology on R is of finite type, then the natural morphism Φ R : R → R** is the pure-injective envelope of R R as a right module over itself.

Finite matrix topologies

A filter F of positive-primitive formulae may be used to give a right R-module M R the structure M F of a topological abelian group. The topology is called a finite matrix topology if every finite matrix subgroup of M R is closed in M F . It is shown that the pure-injective envelope is functorial on the subcategory of modules for which M F is dense in its pure-injective envelope. We call a right R-module almost pure-injective if there is a filter F with respect to which the topological abelian group M F is dense in its pure-injective envelope [ PE ( M ) ] F . In that case, every R-endomorphism of PE ( M R ) is determined by its restriction to M R . When M = R R , this gives the pure-injective envelope PE ( R R ) a ring structure extending that of R, and the proof of this result suggests that this ring is the pure variation of the ring of quotients of a nonsingular ring.

Completion of Modules over Serial, Right Noetherian Rings

Linear topology defined on an arbitrary right module over a right Noetherian serial ring R enables one to describe the reduced, pure injective R-modules as modules that are complete in this topology. With the use of the completion of modules, the pure injective envelope of any right R-module is constructed. Bibliography: 8 titles.

PURE-INJECTIVE ENVELOPES

It is proved that if C is a locally finitely presented additive category, then every object X of C admits a pure-injective envelope. Considered as a morphism, the pure-injective envelope is a pure-monomorphism.

ON MODULES WHICH ARE SUBISOMORPHIC TO THEIR PURE-INJECTIVE ENVELOPES

In the present paper, modules which are subisomorphic (in the sense of Goldie) to their pure-injective envelopes are studied. These modules will be called almost pure-injective modules. It is shown that every module is isomorphic to a direct summand of an almost pure-injective module. We prove that these modules are ker-injective (in the sense of Birkenmeier) over pure-embeddings. For a coherent ring R, the class of almost pure-injective modules coincides with the class of ker-injective modules if and only if R is regular. Generally, the class of almost pure-injective modules is neither closed under direct sums nor under elementary equivalence. On the other hand, it is closed under direct products and if the ring has pure global dimension less than or equal to one, it is closed under reduced products. Finally, pure-semisimple rings are characterized, in terms of almost pure-injective modules.

Pure Injective Envelopes of Finite Length Modules over a Generalized Weyl Algebra

We investigate certain pure injective modules over generalised Weyl algebras. We consider pure injective hulls of finite length modules, the elementary duals of these, torsionfree pure injective modules, and the closure in the Ziegler spectrum of the category of finite length modules supported on a nondegenerate orbit of a generalized Weyl algebra. We also show that this category is a direct sum of uniserial categories and admits almost split sequences. We find parallels to but also marked contrasts with the behaviour of pure injective modules over finite-dimensional algebras and hereditary orders.

On p-injective rings

Throughout this paper R will be an associative ring with unity and all R-modules are unitary. The right (resp. left) annihilator in R of a subset X of a module is denoted by r(X)(resp. I(X)). The Jacobson radical of R is denoted by J(R), the singular ideals are denoted by Z(RR) and Z(RR) and the socles by Soc(RR) and Soc(RR). For a module M, E(M) and PE(M) denote the injective and pure-injective envelopes of M, respectively. For a submodule A ⊆ M, the notation A ⊆⊕M will mean that A is a direct summand of M.

Projective rank functions compatible with pure-injective envelopes

Projective rank functions compatible with pure-injective envelopes

PURE-INJECTIVE ENVELOPE OF A COMMUTATIVE RING AND LOCALIZATIONS

Journal Article PURE-INJECTIVE ENVELOPE OF A COMMUTATIVE RING AND LOCALIZATIONS Get access ALBERTO FACCHINI ALBERTO FACCHINI Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 39, Issue 3, September 1988, Pages 307–321, https://doi.org/10.1093/qmath/39.3.307 Published: 01 September 1988 Article history Received: 10 March 1987 Published: 01 September 1988