Flat R-module Research Articles

On w-copure projective modules

Let R be a commutative ring. An R-module M is said to be w-split if Ext R 1 ( M , N ) is a GV-torsion R-module for all R-modules N. It is known that every projective module is w-split, but the converse is not true in general. In this paper, we study the w-split dimension of a flat module. To do so, we introduce and study the so-called w-copure (resp., strongly w-copure) projective modules, which is in some way a generalization of the notion of copure (resp., strongly copure) projective modules. An R-module M is said to be w-copure projective (resp., strongly w-copure projective) if Ext R 1 ( M , N ) (resp., Ext R n ( M , N ) ) is a GV-torsion R-module for all flat R-modules N and any n ≥ 1 .

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.

Countably generated flat modules are quite flat

We prove that if R is a commutative Noetherian ring, then every countably generated flat R-module is quite flat, i.e., a direct summand of a transfinite extension of localizations of R in countable multiplicative subsets. We also show that if the spectrum of R is of cardinality less than κ, where κ is an uncountable regular cardinal, then every flat R-module is a transfinite extension of flat modules with less than κ generators. This provides an alternative proof of the fact that over a commutative Noetherian ring with countable spectrum, all flat modules are quite flat. More generally, we say that a commutative ring is CFQ if every countably presented flat R-module is quite flat. We show that all von Neumann regular rings and all S-almost perfect rings are CFQ. A zero-dimensional local ring is CFQ if and only if it is perfect. A domain is CFQ if and only if all its proper quotient rings are CFQ. A valuation domain is CFQ if and only if it is strongly discrete.

Flat commutative ring epimorphisms of almost Krull dimension zero

In this paper, we consider flat epimorphisms of commutative rings [Formula: see text] such that, for every ideal [Formula: see text] for which [Formula: see text], the quotient ring [Formula: see text] is semilocal of Krull dimension zero. Under these assumptions, we show that the projective dimension of the R-module U does not exceed 1. We also describe the Geigle–Lenzing perpendicular subcategory [Formula: see text] in [Formula: see text]. Assuming additionally that the ring U and all the rings [Formula: see text] are perfect, we show that all flat R-modules are U-strongly flat. Thus, we obtain a generalization of some results of the paper [ 6 ], where the case of the localization [Formula: see text] of the ring R at a multiplicative subset [Formula: see text] was considered.

On Rings of Weak Global Dimension at Most One

A ring R is of weak global dimension at most one if all submodules of flat R-modules are flat. A ring R is said to be arithmetical (resp., right distributive or left distributive) if the lattice of two-sided ideals (resp., right ideals or left ideals) of R is distributive. Jensen has proved earlier that a commutative ring R is a ring of weak global dimension at most one if and only if R is an arithmetical semiprime ring. A ring R is said to be centrally essential if either R is commutative or, for every noncentral element x∈R, there exist two nonzero central elements y,z∈R with xy=z. In Theorem 2 of our paper, we prove that a centrally essential ring R is of weak global dimension at most one if and only is R is a right or left distributive semiprime ring. We give examples that Theorem 2 is not true for arbitrary rings.

The Homotopy Category of Cotorsion Flat Modules

This paper aims at studying the homotopy category of cotorsion flat left modules $${{\mathbb {K}}({\mathrm{CotF}}\text {-}R)}$$ over a ring R. We prove that if R is right coherent, then the homotopy category $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)$$ of dg-cotorsion complexes of flat R-modules is compactly generated. This uses firstly the existence of cotorsion flat preenvelopes over such rings and, secondly, the existence of a complete cotorsion pair $$({{\mathbb {K}}_{\mathrm{p}}({\mathrm{Flat}}\text {-}R)}, {\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R))$$ in the homotopy category $${{\mathbb {K}}({\mathrm{Flat}}\text {-}R)}$$ of complexes of flat R-modules, for arbitrary R. In the setting of quasi coherent sheaves over a Noetherian scheme, this cotorsion pair was discovered in the literature. However, we use a more elementary argument that gives this cotorsion pair for arbitrary R. Next we deal with cotorsion flat resolutions of complexes and define and study the notion of cotorsion flat dimension for complexes of flat R-modules. We also obtain an equivalence $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)\approx {{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ of triangulated categories where $${{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ is the homotopy category of projective R-modules. Combined with the aforementioned result, this recovers a result from Neeman, asserting the compact generation of $${{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ over right coherent R. Also we get the unbounded derived category $${\mathbb {D}} (R)$$ of R as a Verdier quotient of $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)$$ .

ON SEMI-n-ABSORBING SUBMODULES

Let R be a commutative ring with identity and n a positive integer greater than 1. In this paper, we introduce the concept of semi-n-absorbing submodules. A proper submodule N of an R-module M is called a semi-n-absorbing submodule of M if whenever a \in R, x \in M and a^nx \in N then ax \in N ora^n \in (N :R M). A number of results concerning semi-n-absorbing submodulesand examples of them are given. It is shown that if N is a semi-n-absorbing submodule of M and F is a flat R-module such that F \otimes N is proper in F \otimes M then F \otimes N is semi-n-absorbing in F \otimes M. We show that the converse holds when F is faithfully flat.

Covering classes, strongly flat modules, and completions

We study some closely interrelated notions of Homological Algebra: (1) We define a topology on modules over a not-necessarily commutative ring R that coincides with the R-topology defined by Matlis when R is commutative. (2) We consider the class $$ \mathcal {SF}$$ of strongly flat modules when R is a right Ore domain with classical right quotient ring Q. Strongly flat modules are flat. The completion of R in its R-topology is a strongly flat R-module. (3) We prove some results related to the question whether $$ \mathcal {SF}$$ a covering class implies $$ \mathcal {SF}$$ closed under direct limits. This is a particular case of the so-called Enochs’ Conjecture (whether covering classes are closed under direct limits). Some of our results concern right chain domains. For instance, we show that if the class of strongly flat modules over a right chain domain R is covering, then R is right invariant. In this case, flat R-modules are strongly flat.

On the vanishing of annihilators of modules

The main purpose of this paper is to generalize a result of T.G. Lucas which states that if R is a reduced commutative ring and M is a flat R-module, then the idealization is an -ring if and only if R is an -ring [14, Proposition 3.5]. In effect, we drop the reduceness hypotheses and prove that, given an arbitrary commutative ring R and any submodule M of a flat R-module F, is an -ring (resp., -ring) if and only if R is an -ring (resp., -ring).

S-almost perfect commutative rings

Given a multiplicative subset S in a commutative ring R, we consider S-weakly cotorsion and S-strongly flat R-modules, and show that all R-modules have S-strongly flat covers if and only if all flat R-modules are S-strongly flat. These equivalent conditions hold if and only if the localization RS is a perfect ring and, for every element s∈S, the quotient ring R/sR is a perfect ring, too. The multiplicative subset S⊂R is allowed to contain zero-divisors.

Duality pairs and stable module categories

Let R be a commutative ring. We show that any complete duality pair gives rise to a theory of relative homological algebra, analogous to Gorenstein homological algebra. Indeed Gorenstein homological algebra over a commutative Noetherian ring of finite Krull dimension can be recovered from the duality pair (F,I) where F is the class of flat R-modules and I is the class of injective R-modules. For a general R, the AC-Gorenstein homological algebra of Bravo–Gillespie–Hovey is the one coming from the duality pair (L,A) where L is the class of level R-modules and A is class of absolutely clean R-modules. Indeed we show here that the work in [1] can be extended to obtain similar abelian model structures on R-Mod from any a complete duality pair (L,A). It applies in particular to the original duality pairs constructed by Holm–Jørgensen.

Relative singularity categories with respect to gorenstein flat modules

Let R be a right coherent ring and D b (R-Mod) the bounded derived category of left R-modules. Denote by \({D^b}{\left( {R - Mod} \right)_{\widehat {\left[ {GF,C} \right]}}}\) the subcategory of D b (R-Mod) consisting of all complexes with both finite Gorenstein flat dimension and cotorsion dimension and K b (F ∩ C) the bounded homotopy category of flat cotorsion left R-modules. We prove that the quotient triangulated category \({D^b}{\left( {R - Mod} \right)_{\widehat {\left[ {GF,C} \right]}}}\)/K b (F ∩ C) is triangle-equivalent to the stable category GF ∩ C of the Frobenius category of all Gorenstein flat and cotorsion left R-modules.

On Balance for Relative Homology

Let R be a ring. It is reasonable to use a strict 𝒲𝒳-resolution of a right R-module M of finite 𝒳-projective dimension, where 𝒳 denotes a subcategory of right R-modules closed under extensions and admits an injective cogenerator 𝒲, to define the relative homology functor . A general balance result is established for such relative homology functor that encompasses a theorem of Emmanouil on Gorenstein projective R-modules and extends a theorem of Holm on Gorenstein flat R-modules. We also consider the above relative homology functor with respect to subcategories arising from an arbitrary but fixed semidualizing R-module. Inspired by the idea of Sather-Wagstaff et al. [25], we obtain the corresponding balance results when R is Cohen–Macaulay with a dualizing R-module as applications of our balance result.

On Stability of F-Gorenstein Flat Categories

To investigate cohomology theories based on flats, Asadollahi and Salarian gave the definition of F-Gorenstein flat R-modules, and these modules are exactly Gorenstein flat provided that R is right coherent. In this paper, we get some properties of F-Gorenstein flat R-modules and establish the stability of F-Gorenstein flat categories.

On n-Coherent Rings and (n,d)-Injective Modules

Let R be a ring, n, d be fixed non-negative integers, [Formula: see text] the class of (n,d)-injective left R-modules, and [Formula: see text] the class of (n,d)-flat right R-modules. In this paper, we prove that if R is a left n-coherent ring and m ≥ 2, then [Formula: see text] if and only if [Formula: see text], if and only if Ext m+k(M,N) = 0 for all left R-modules M, N and all k ≥ -1, if and only if Ext m-1(M,N) = 0 for all left R-modules M, N. Meanwhile, we prove that if R is a left n-coherent ring, then − ⊗ − is right balanced on [Formula: see text] by [Formula: see text], and investigate the global right [Formula: see text]-dimension of [Formula: see text] and the global right [Formula: see text]-dimension of [Formula: see text] by right derived functors of − ⊗ −. Some known results are obtained as corollaries.

Homological Dimensions with Respect to a Semidualizing Module and Tensor Products of Algebras

Let C be a semidualizing module for a commutative ring R. It is shown that the [Formula: see text]-injective dimension has the ability to detect the regularity of R as well as the [Formula: see text]-projective dimension. It is proved that if D is dualizing for a Noetherian ring R such that id R(D) = n < ∞, then [Formula: see text] for every flat R-module F. This extends the result due to Enochs and Jenda. Finally, over a Noetherian ring R, it is shown that if M is a pure submodule of an R-module N, then [Formula: see text]. This generalizes the result of Enochs and Holm.

Notes on Gorenstein cotorsion modules

Let R be a ring. An R-module N is called Gorenstein cotorsion module if ExtR1 (F, N) = 0 for any Gorenstein flat R-module F. We investigate some properties of Gorenstein cotorsion envelopes and introduce Gorenstein cotorsion dimension. Some characterizations of Gorenstein cotorsion dimension of modules and rings are given.

A Gorenstein analogue of a result of Bertin

An integral domain R is called a Gorenstein Prüfer (G-Prüfer) domain if it is coherent and every submodule of a flat R-module is Gorenstein flat. In this paper, we show that every n-FC domain is an intersection of local G-Prüfer domains. We also give several characterizations of G-Prüfer domains.

Σ-semi-compact rings and modules

In this paper, several characterizations of semi-compact modules are given. Among other results, we study rings whose semi-compact modules are injective. We introduce the property Σ-semi-compact for modules and we characterize the modules satisfying this property. In particular, we show that a ring R is left Σ-semi-compact if and only if R satisfies the ascending (respectively, descending) chain condition on the left (respectively, right) annulets. Moreover, we prove that every flat left R-module is semi-compact if and only if R is left Σ-semi-compact. We also show that a ring R is left Noetherian if and only if every pure projective left R-module is semi-compact. Finally, we consider rings whose flat modules are finitely (singly) projective. For any commutative arithmetical ring R with quotient ring Q, we prove that every flat R-module is semi-compact if and only if every flat R-module is finitely (singly) projective if and only if Q is pure semisimple. A similar result is obtained for reduced commutative rings R with the space Min R compact. We also prove that every (ℵ0, 1)-flat left R-module is singly projective if R is left Σ-semi-compact, and the converse holds if Rℕis an (ℵ0, 1)-flat left R-module.

Schwach-Flache Moduln

Let R be a commutative, noetherian ring, and let M be an R-module. M is called weakly flat if the kernel of any epimorphism Y ↠ M is closed in Y. Equivalently, a consequence of M/U being singular is that U is essential in M, or that Ann R(𝔭)·M is essential in Ann M(𝔭) for all 𝔭 ∈AssR(M). For a series of important module classes we give an explicit description of their weakly flat objects, and in the local case we investigate the connection between the weak flatness of M and the weak injectivity of the Matlis dual M°. Finally, we characterize those rings R over which every weakly flat R-module is already flat.