FP-injective Modules Research Articles

Triviality criteria for unbounded complexes

We study properties of modules that appear as syzygies of acyclic complexes of projective, injective, flat or flat-cotorsion modules and obtain criteria for these complexes to be contractible or totally acyclic. Our results illustrate the importance of strongly fp-injective modules in the study of these properties. We examine implications of the existence of complete resolutions (in a certain weak sense) and the finiteness of the Gorenstein projective dimension of pure-projective modules. We also use the orthogonality in the homotopy category between complexes of flat modules and pure acyclic complexes of cotorsion modules, in order to study the syzygies of acyclic complexes of flat modules. Finally, we present some applications of our results to group rings, regarding complete resolutions and cohomological periodicity.

On K-absolutely pure complexes

In this paper, we examine the class of K-absolutely pure complexes. These are the complexes which are right orthogonal in the homotopy category K(R) to the acyclic complexes of pure-projective modules. The class K-abspure of these complexes is preenveloping in K(R); in fact, a Bousfield localization exists for the embedding K-abspure⊆K(R) and the quotient K(R)/K-abspure is equivalent to the homotopy category of acyclic complexes of pure-projective modules. We examine the role of K-absolutely pure complexes in the pure derived category Dpure(R) and show that K-abspure is the isomorphic closure of the class of K-injective complexes therein. We explore the relevance of strongly fp-injective modules in the study of K-absolutely pure complexes and characterize the K-absolutely pure complexes of strongly fp-injective modules. Finally, we show that a K-absolutely pure complex of strongly fp-injective modules admits a K-injective complex of injective modules as a K(PInj)-preenvelope, in the case where the ring is left coherent. The notion of K-absolute purity is dual to the notion of K-flatness in the homotopy category, in a way analogous to the duality between (strongly) fp-injectivity and flatness in the module category.

Weakly Ding injective modules and complexes

We consider two classes of modules over coherent rings: the Gorenstein FP-injective modules, and the weakly Ding injective modules. The Gorenstein FP-injective modules are the cycles of the exact complexes of FP-injective modules. The weakly Ding injective modules are the cycles of the exact complexes of FP-injective modules that stay exact when applying a functor for any FP-injective module A. We prove that the class of Gorenstein FP-injective modules is both covering and preenveloping over any (left) coherent ring with the property that every injective module has finite flat dimension. We also prove that, over the same type of rings, the class of weakly Ding injectives, , is preenveloping in . If, moreover, is closed under extensions, then is a hereditary cotorsion pair. In particular, this is the case when R is a Ding Chen ring, and is closed under extensions. We show that if R is a Ding Chen ring such that every FP-injective module has finite projective dimension, then the two classes of modules coincide. Consequently, the class of weakly Ding injective modules is also covering in this case. In the last part of the paper we prove some analogue results for Gorenstein FP-injective complexes and for weakly Ding injective complexes.

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.

On subinjectivity domains of RD-injective modules

The subjectivity domain of a module allows us to evaluate how close (or far) a module is from being injective. In this paper, we consider three families of rings characterized by their RD-injective left [Formula: see text]-modules via the subinjective domain: those whose noninjective RD-injective left [Formula: see text]-modules are subinjective relatively only to divisible modules, those whose noninjective RD-injective left [Formula: see text]-modules are subinjective relatively only to fp-injective modules and those whose noninjective RD-injective left [Formula: see text]-modules are subinjective relatively only to injective modules.

Relative FP-injective modules and relative IF rings

Let R be a commutative ring and w be the so-called w-operation on R. Then we introduce and study two concepts of w-FP-injective modules and w-IF rings. To do so, we use two main methods, of which one is to localize at maximal w-ideals of R and the other is to utilize w-Nagata modules over w-Nagata rings. As an application, we characterize Prüfer v-multiplication domains in terms of w-FP-injective modules. Finally we provide an example of a w-IF ring, but not an IF ring using a trivial extension.

Acyclic Complexes and Gorenstein Rings

For a given class of modules [Formula: see text], let [Formula: see text] be the class of exact complexes having all cycles in [Formula: see text], and dw([Formula: see text]) the class of complexes with all components in [Formula: see text]. Denote by [Formula: see text][Formula: see text] the class of Gorenstein injective R-modules. We prove that the following are equivalent over any ring R: every exact complex of injective modules is totally acyclic; every exact complex of Gorenstein injective modules is in [Formula: see text]; every complex in dw([Formula: see text][Formula: see text]) is dg-Gorenstein injective. The analogous result for complexes of flat and Gorenstein flat modules also holds over arbitrary rings. If the ring is n-perfect for some integer n ≥ 0, the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules. We also prove the following characterization of Gorenstein rings. Let R be a commutative coherent ring; then the following are equivalent: (1) every exact complex of FP-injective modules has all its cycles Ding injective modules; (2) every exact complex of flat modules is F-totally acyclic, and every R-module M such that M+ is Gorenstein flat is Ding injective; (3) every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that M+ is Gorenstein flat is Ding injective. If R has finite Krull dimension, statements (1)–(3) are equivalent to (4) R is a Gorenstein ring (in the sense of Iwanaga).

Some characterizations of coherent rings in terms of strongly FP-injective modules

An R-module M is called strongly FP-injective if for any finitely presented R-module P and any i > 0. Denoted by the class of all strongly FP-injective R-modules and by the left orthogonal class with respect to Comparing with some classical results of Noetherian rings, we show that a ring is coherent if and only if is closed under pure submodules; if and only if any finitely generated module in is finitely presented; if and only if R is -coherent and is closed under direct sums; if and only if for any nonnegative integer n, any finitely presented left R-module X, any R-T-bimodule M and any injective right T-module E. In addition, we show that a module is injective if and only if it is a pure -periodic module, where denotes the class of all injective modules.

Pure-minimal chain complexes

We introduce a notion of pure-minimality for chain complexes of modules and show that it coincides with (homotopic) minimality in standard settings, while being a more useful notion for complexes of flat modules. As applications, we characterize von Neumann regular rings and left perfect rings.

Triangle equivalences involving Gorenstein FP-injective modules

ABSTRACTRecently, Hu et al. [16] introduced the notion of Gorenstein FP-injective modules and investigated a notion of Gorenstein FP-injective dimension for complexes. Let R be a left coherent ring and Db(R-Mod) the bounded derived category of left R-modules. It is proved that the quotient triangulated category of the subcategory of Db(R-Mod) consisting of complexes with both finite Gorenstein FP-injective dimension and FP-projective dimension by the bounded homotopy category of FP-projective-injective left R-modules is triangle-equivalent to the stable category of the Frobenius category of all Gorenstein FP-injective and FP-projective left R-modules. Note that a similar argument is also valid for the case of Gorenstein injective left R-modules. We extend a triangle equivalence established by Beligiannis involving Gorenstein injective left R-modules from rings with finite left Gorenstein global dimension to arbitrary rings.

On the Structure of Modules Defined by Opposites of FP Injectivity

Let R be a ring with unity and let $$M_R$$ and $$_RN$$ be right and left modules,respectively. The module $$M_R$$ is said to be absolutely $$_RN$$ -pure if $$M \otimes N \rightarrow L \otimes N$$ is amonomorphism for every extension $$L_R$$ of $$M_R$$ . For a module $$M_R$$ , the subpurity domain of $$M_R$$ is defined to be the collection of all modules $$_RN$$ , such that $$M_R$$ is absolutely $$_RN$$ -pure. Clearly, $$M_R$$ is absolutely $$_RF$$ -pure for every flat module $$_RF$$ and that $$M_R$$ is FP-injective if the subpurity domain of M is the entire class of left modules. As an opposite of FP-injective modules, $$M_R$$ is said to be a test for flatness by subpurity (or t.f.b.s. for short) if its subpurity domain is as small as possible, namely, consisting of exactly the flat left modules. We characterize the structure of t.f.b.s. modules over commutative hereditary Noetherian rings. We prove that a module M is t.f.b.s. over a commutative hereditary Noetherian ring if and only if M / Z(M) is t.f.b.s. if and only if $${\text {Hom}}(M/Z(M), S) \ne 0$$ for each singular simple module S. Prufer domains are characterized as those domains all of whose nonzero finitely generated ideals are t.f.b.s.

Gorenstein coresolving categories

ABSTRACTLet 𝒜 be an abelian category. A subcategory 𝒳 of 𝒜 is called coresolving if 𝒳 is closed under extensions and cokernels of monomorphisms and contains all injective objects of 𝒜. In this paper, we introduce and study Gorenstein coresolving categories, which unify the following notions: Gorenstein injective modules [8], Gorenstein FP-injective modules [20], Gorenstein AC-injective modules [3], and so on. Then we define a resolution dimension relative to the Gorenstein coresolving category 𝒢ℐ𝒳(𝒜). We investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, we study stability of the Gorenstein coresolving category 𝒢ℐ𝒳(𝒜) and apply the obtained properties to special subcategories and in particular to module categories.

CARTAN–EILENBERG FP-INJECTIVE COMPLEXES

In this article, we extend the notion of FP-injective modules to that of Cartan–Eilenberg complexes. We show that a complex $C$ is Cartan–Eilenberg FP-injective if and only if $C$ and $\text{Z}(C)$ are complexes consisting of FP-injective modules over right coherent rings. As an application, coherent rings are characterized in various ways, using Cartan–Eilenberg FP-injective and Cartan–Eilenberg flat complexes.

Strongly FP-injective modules

ABSTRACTA module M is called strongly FP-injective if Exti(P,M) = 0 for any finitely presented module P and all i≥1. (Pre)envelopes and (pre)covers by strongly FP-injective modules are studied. We also use these modules to characterize coherent rings. An example is given to show that (strongly) FP-injective (pre)covers may fail to be exist in general. We also give an example of a module that is FP-injective but not strongly FP-injective.

Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre–Grothendieck duality

For a left coherent ring A with every left ideal having a countable set of generators, we show that the coderived category of left A-modules is compactly generated by the bounded derived category of finitely presented left A-modules (reproducing a particular case of a recent result of Stovicek with our methods). Furthermore, we present the definition of a dualizing complex of fp-injective modules over a pair of noncommutative coherent rings A and B, and construct an equivalence between the coderived category of A-modules and the contraderived category of B-modules. Finally, we define the notion of a relative dualizing complex of bimodules for a pair of noncommutative ring homomorphisms A \to R and B \to S, and obtain an equivalence between the R/A-semicoderived category of R-modules and the S/B-semicontraderived category of S-modules. For a homomorphism of commutative rings A\to R, we also construct a tensor structure on the R/A-semicoderived category of R-modules. A vision of semi-infinite algebraic geometry is discussed in the introduction.

An alternative perspective on flatness of modules

Given modules [Formula: see text] and [Formula: see text], [Formula: see text] is said to be absolutely [Formula: see text]-pure if [Formula: see text] is a monomorphism for every extension [Formula: see text] of [Formula: see text]. For a module [Formula: see text], the absolutely pure domain of [Formula: see text] is defined to be the collection of all modules [Formula: see text] such that [Formula: see text] is absolutely [Formula: see text]-pure. As an opposite to flatness, a module [Formula: see text] is said to be f-indigent if its absolutely pure domain is smallest possible, namely, consisting of exactly the fp-injective modules. Properties of absolutely pure domains and off-indigent modules are studied. In particular, the existence of f-indigent modules is determined for an arbitrary rings. For various classes of modules (such as finitely generated, simple, singular), necessary and sufficient conditions for the existence of f-indigent modules of those types are studied. Furthermore, f-indigent modules on commutative Noetherian hereditary rings are characterized.

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.

Gorenstein categories $\mathcal G(\mathscr X,\mathscr Y,\mathscr Z)$ and dimensions

Let $\mathscr {A}$ be an abelian category and $\mathscr {X},\mathscr {Y},\mathscr {Z}$ additive full subcategories of $\mathscr {A}$. We introduce and study the Gorenstein category $\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})$ as a common generalization of some known modules such as Gorenstein projective (injective) modules \cite {EJ95}, strongly Gorenstein flat modules \cite {DLM} and Gorenstein FP-injective modules \cite {DM}, and prove the stability of $\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})$. We also establish Gorenstein homological dimensions in terms of the category $\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})$.

<italic>n</italic>- 强Gorenstein FP- 内射模

For any positive integer <italic>n</italic>, we introduce and study <italic>n</italic>-strongly Gorenstein FP-injective modules. Some examples are given to show that <italic>n</italic>-strongly Gorenstein FP-injective modules need not be <italic>m</italic>-strongly Gorenstein FP-injective modules whenever <italic>n</italic> > <italic>m</italic>. Many properties of <italic>n</italic>-strongly Gorenstein FP-injective modules are discussed, some known results are obtained as corollaries. Finally, we characterize <italic>n</italic>-strongly Gorenstein Von Neumann regular rings in terms of <italic>n</italic>-strongly Gorenstein FP-injective modules.

On Strongly Gorenstein FP-Injective Modules

This article continues to investigate a particular case of Gorenstein FP-injective modules, called strongly Gorenstein FP-injective modules. Some examples are given to show that strongly Gorenstein FP-injective modules lie strictly between FP-injective modules and Gorenstein FP-injective modules. Various results are developed, many extending known results in [1]. We also characterize FC rings in terms of strongly Gorenstein FP-injective, projective, and flat modules.