Left Noetherian Ring Research Articles

Auslander conditions and tilting-like cotorsion pairs

We study homological behavior of modules satisfying the Auslander condition. Assume that AC is the class of left R-modules satisfying the Auslander condition. It is proved that each cycle of an exact complex with each term in AC belongs to AC for any ring R. As a consequence, we show that for any left Noetherian ring R, AC is a resolving subcategory of the category of left R-modules if and only if RR satisfies the Auslander condition if and only if each Gorenstein projective left R-module belongs to AC. As an application, we prove that, for an Artinian algebra R satisfying the Auslander condition, R is Gorenstein if and only if AC coincides with the class of Gorenstein projective left R-modules if and only if (AC<∞,(AC<∞)⊥) is a tilting-like cotorsion pair if and only if (AC<∞,I) is a tilting-like cotorsion pair, where AC<∞ is the class of left R-modules with finite AC-dimension and I is the class of injective left R-modules. This leads to some criteria for the validity of the Auslander and Reiten conjecture which says that an Artinian algebra satisfying the Auslander condition is Gorenstein.

$${{\,\mathrm{\texttt {VIC}}\,}}$$-modules over noncommutative rings

For a finite ring $R$, not necessarily commutative, we prove that the category of $\text{VIC}(R)$-modules over a left Noetherian ring $\mathbf{k}$ is locally Noetherian, generalizing a theorem of the authors that dealt with commutative $R$. As an application, we prove a very general twisted homology stability for $\text{GL}_n(R)$ with $R$ a finite noncommutative ring.

On annihilator ideals in skew power series rings

The purpose of this paper is to study the behavior of the IN and SA property with respect to the skew power series and Laurent series extensions. For an α-compatible ring R it is shown that the skew power series ring is right SA if and only if R is right SA and satisfies the (SQA2) condition. Also it is shown that a semiprime ring R is right SA if and only if is right SA. Moreover, if α is an automorphism, then we have similar results for skew Laurent series ring Furthermore, if R is a reduced left Noetherian ring, then R is left IN if and only if is left IN if and only if is left IN. Also we present some examples to show that if we relax each assumption on R, our results can fail.

Unbounded derived categories of small and big modules: Is the natural functor fully faithful?

Consider the obvious functor from the unbounded derived category of all finitely generated modules over a left noetherian ring $R$ to the unbounded derived category of all modules. We answer the natural question whether this functor defines an equivalence onto the full subcategory of complexes with finitely generated cohomology modules in two special cases. If $R$ is a quasi-Frobenius ring of infinite global dimension, then this functor is not full. If $R$ has finite left global dimension, this functor is an equivalence. We also prove variants of the latter assertion for left coherent rings, for noetherian schemes and for locally noetherian Grothendieck categories.

On the Restricted Minimum Condition for Rings

Generalizing Artinian rings, a ring R is said to have right restricted minimum condition ( $${\mathrm{r.RMC}}$$ , for short) if R/A is an Artinian right R-module for any essential right ideal A of R. It is asked in Jain et al. [Cyclic Modules and the Structure of Rings, Oxford University Press, Oxford, 2012, 3.17 Questions (2)] that (i) Is a left self-injective ring with $${\mathrm{r.RMC}}$$ quasi-Frobenius? (ii) Whether a serial ring with $${\mathrm{r.RMC}}$$ must be Noetherian? We carry out a study of rings with $${\mathrm{r.RMC}}$$ and determine when a right extending ring has $${\mathrm{r.RMC}}$$ in terms of rings $${\begin{bmatrix} S&{}\quad M\\ 0&{}\quad R\end{bmatrix}}$$ such that S is right Artinian, $$M_{Q}$$ is semisimple ( $$Q={\mathrm{Q}}(R)$$ ) and R is a semiprime ring with Krull dimension 1. We proved that a left self-injective ring R with $${\mathrm{r.RMC}}$$ is quasi-Frobenius if and only if $$\hbox {Z}_{r}(R) = \hbox {Z}_{l}(R)$$ if and only if $$\hbox {Z}_{r}(R)$$ is a finitely generated left ideal and $${\mathrm{N}}(R)\cap {\mathrm{Soc}}(R_{R})$$ is a finitely generated right ideal. Right serial rings with $${\mathrm{r.RMC}}$$ are studied and proved that a non-singular serial ring has $${\mathrm{r.RMC}}$$ if and only if it is a left Noetherian ring. Examples are presented to describe our results and to show that $$\mathrm{RMC}$$ is not symmetric for a ring.

Rings described by some special morphisms of modules

The category of morphisms of R-modules is an expansion of the category of R-modules. Using some special morphisms of modules such as phantom, -phantom, flat and absolutely pure morphisms, we give some new characterizations of left coherent rings, von Neumann regular rings, left semihereditary rings, left Noetherian rings, and right perfect rings.

Sofic mean length

Given a length function L on the R-modules of a unital ring R, for each sofic group Γ we define a mean length for every locally L-finite RΓ-module relative to a bigger RΓ-module. We establish an addition formula for the mean length.We give two applications. The first one shows that for any unital left Noetherian ring R, RΓ is stably direct finite. The second one shows that for any ZΓ-module M, the mean topological dimension of the induced Γ-action on the Pontryagin dual of M coincides with the von Neumann–Lück rank of M.

The Socle of the Last Term in the Minimal Injective Resolution of a Gorenstein Module

Let $R$ be a left Noetherian ring, $S$ a right Noetherian ring and $_RU$ a Gorenstein module with $S={\rm End}(_RU)$. If the injective dimensions of $_RU$ and $U_S$ are finite, then the last term in the minimal injective resolution of $_{R}U$ has an essential socle.

Strongly &lt;i&gt;Ƥ&lt;/i&gt;-projective Modules and &lt;i&gt;Ƥ&lt;/i&gt;-projective Complexes

In this paper we first study the properties of Strongly Ƥ-projective modules, and obtain some equivalent conclusions about Strongly Ƥ-projective modules, it is proved that a finitely generated right R-module N is strongly Ƥ-projective if and only if Exti (N, R) = 0 for all i ≥ 1 over left noetherian and right perfect ring, a Ƥ-projective right R-module N is strongly Ƥ-projective if and only if the first syzygy of N is strongly Ƥ-projective. Then we extend the notion of Ƥ-projective modules to that of Ƥ-projective complexes. We study the relationships between Ƥ-projective complexes and Ƥ-projective modules, it is proved that a complex C is Ƥ-projective if and only if every Ci is Ƥ-projective for every integer i if and only if Ext1 (C, P) = 0 for every projective complex Ƥ if and only if for every exact sequence 0→A→Ƥ→C→0 with Ƥ projective, A→Ƥ is a projective preenvelope of A. Some characterizations of Ƥ-projective complexes also obtained.

Singular equivalences of commutative noetherian rings and reconstruction of singular loci

Two left noetherian rings R and S are said to be singularly equivalent if their singularity categories are equivalent as triangulated categories. The aim of this paper is to give a necessary condition for two commutative noetherian rings to be singularly equivalent. To do this, we develop a support theory for triangulated categories without tensor structure.

A note on Gorenstein projective modules

ABSTRACTIn this note, it is proven that if R is a right noetherian ring with and for any left R-module F with finite flat dimension, then M is Gorenstein projective; if R is a left noetherian ring with idRR<∞ and M is a Gorenstein projective left R-module, then for any left R-module F with finite flat dimension.

Criteria for a Ring to have aLeft Noetherian Largest Left Quotient Ring

Criteria are given for a ring to have a left Noetherian largest left quotient ring. It is proved that each such a ring has only finitely many maximal left denominator sets. An explicit description of them is given. In particular, every left Noetherian ring has only finitely many maximal left denominator sets.

The transpose of modules relative to subcategories

Let R be a left noetherian ring and S a right noetherian ring, and let X be a subcategory of finitely generated left R-modules and BSR a finite (R,S)-bimodule. As a generalization of the Auslander transpose, replacing a projective presentation by an X-presentation and the functor HomR(−,R) by the functor HomR(−,B), we introduce the X(B)-transpose of an R-module M admitting an X-presentation. For a suitable subcategory X, some useful properties of X(B)-transposes are obtained. It is shown that the X(B)-transpose shares many nice properties with the Auslander transpose. Some known results are obtained as corollaries.

Strongly copure projective, injective and flat complexes

In this paper, we extend the notions of strongly copure projective, injective and flat modules to that of complexes and characterize these complexes. We show that the strongly copure projective precover of any finitely presented complex exists over $n$-FC rings, and a strongly copure injective envelope exists over left Noetherian rings. We prove that strongly copure flat covers exist over arbitrary rings and that $(\mathcal {SCF},\mathcal {SCF}^\bot )$ is a perfect hereditary cotorsion theory where $\mathcal {SCF}$ is the class of strongly copure flat complexes.

A note on Gorenstein transposes

Let [Formula: see text] be a left and right Noetherian ring. In this paper, we prove that any Gorenstein transpose of a finitely generated [Formula: see text]-module is exactly an Auslander transpose. As applications, we obtain a new relation between a Gorenstein transpose of a module with a transpose of the same module, and show that the Gorenstein transpose of a module is unique up to Gorenstein projective equivalence. In addition, when [Formula: see text] is an Artin algebra, the corresponding Auslander–Reiten sequences are constructed in terms of Gorenstein transposes.

Weakly left localizable rings

ABSTRACTA new class of rings, the class of weakly left localizable rings, is introduced. A ring R is called weakly left localizable if each non-nilpotent element of R is invertible in some left localization S−1R of the ring R. Explicit criteria are given for a ring to be a weakly left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case for all left Noetherian rings). It is proved that a ring with finitely many maximal left denominator sets that satisfies some natural conditions is a weakly left localizable ring iff its left quotient ring is a direct product of finitely many local rings such that their radicals are nil ideals.

Stacked Decomposition Theorem for Modules Over Serial Left Noetherian Rings

A theorem on the stacked decomposition for infinitely generated projective left modules over serial left noetherian rings is proved.

HOMOLOGICAL DIMENSIONS OF CROSSED PRODUCTS

AbstractIn this paper, we consider several homological dimensions of crossed products AασG, where A is a left Noetherian ring and G is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of AσαG are classified: global dimension of AσαG is either infinity or equal to that of A, and finitistic dimension of AσαG coincides with that of A. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that A is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow p-subgroup S ≤ G, we show that A and AασG share the same homological dimensions under extra assumptions, extending the main results in (Li, Representations of modular skew group algebras, Trans. Amer. Math. Soc.367(9) (2015), 6293–6314, Li, Finitistic dimensions and picewise hereditary property of skew group algebras, to Glasgow Math. J.57(3) (2015), 509–517).

Artinian Serial Modules over Commutative (or, Left Noetherian) Rings are at Most One Step Away from Being Noetherian

We introduce and study the concept of dual perfect dimension which is a Krull-like dimension extension of the concept of acc on finitely generated submodules. We observe some basic facts for modules with this dimension, which are similar to the basic properties of modules with Noetherian dimension. For Artinian serial modules, we show that these two dimensions coincide. Consequently, we prove that the Noetherian dimension of non-Noetherian Artinian serial modules over the rings of the title is 1.

On Maximal Torsion Radicals, IV

It is well-known that if R is a left Noetherian ring, then there is a bijective correspondence between minimal prime ideals of R and maximal torsion radicals of R-Mod. Using the notion of a prime M-ideal, it is shown that this correspondence can be extended to the category σ[M] of modules subgenerated by a module M, provided that M is a Noetherian quasi-projective generator in σ[M]. Furthermore, under this hypothesis the prime M-ideals are the fully invariant submodules P of M such that M/P is semi-compressible.