A Class of Auslander-Regular Macaulay Rings

Abstract

For a (left and right) noetherian semilocal ring R we analyse a regularity concept (called weak regularity) based on the equation gld R = dim R. Examples are regular Cohen-Macaulay orders over a regular local ring, localized enveloping algebras of finite dimensional Lie algebras, and the regular rings classified in Rump (2001b). We prove that weakly regular rings are Auslander-regular and Macaulay.     

The Converse of Schur's Lemma in Noetherian Rings and Group Algebras

ABSTRACT

If M is a simple module over a ring R, then, by Schur's Lemma, its endomorphism ring is a division ring. However, the converse of this property, which we called the CSL property, does not hold in general. The object of this article is to study this converse for a few classes of rings: left Noetherian rings, V-rings and group algebras. First, we establish that a left Noetherian ring R is a CSL ring if and only if a ring R is left–artinian and primary decomposable. Secondly, we prove that a left semiartinian V-ring is CSL. At last, we study the CSL property in group algebra K [ G ] where K a field algebraically closed of characteristic p and G is a finite group of order divisible by p. Our main contribution is that K [ G ] is a CSL ring if and only if Gbf = HP where H is a normal p′-subgroup and bfP a Sylow bfp-subgroup of bfG. In this case, K [ G ] is primary decomposable.     

Good Magma Gradings on Rings

Suppose that G and H are magmas and that R is a strongly G-graded ring. We show that there is a bijection between the set of good (zero) H-gradings of R and the set of (zero) magma homomorphisms from G to H. Thereby we generalize a result by D?sc?lescu, Ion, N?st?sescu, and Rios Montes from group gradings of matrix rings to strongly magma graded rings. We also show that there is an isomorphism between the preordered set of good (zero) H-filters on R and the preordered set of (zero) submagmas of G × H. These results are applied to category graded rings and, in particular, to the case when G and H are groupoids. In the latter case, we use this bijection to determine the cardinality of the set of good H-gradings on R.     

The generating hypothesis in the derived category of a ring

We show that a strong form (the fully faithful version) of the generating hypothesis, introduced by Freyd in algebraic topology, holds in the derived category of a ring R if and only if R is von Neumann regular. This extends results of the second author (J. Pure Appl. Algebra 208(2), 2007). We also characterize rings for which the original form (the faithful version) of the generating hypothesis holds in the derived category of R. These must be close to von Neumann regular in a precise sense, and, given any of a number of finiteness hypotheses, must be von Neumann regular. However, we construct an example of such a ring that is not von Neumann regular and therefore does not satisfy the strong form of the generating hypothesis.     

The Krull–Gabriel Dimension of a Serial Ring

Abstract

We prove that every serial ring R has the isolation property: every isolated point in any theory of modules over R is isolated by a minimal pair. Using this we calculate the Krull–Gabriel dimension of the module category over serial rings. For instance, we show that this dimension cannot be equal to 1.     

Ideal-comparability over Regular Rings

We introduce the concept of ideal-comparability condition for regular rings. Let I be an ideal of a regular ring R. If R satisfies the I-comparability condition, then R is one-sided unit-regular if and only if so is R/I. Also, we show that a regular ring R satisfies the general comparability if and only if the following hold： （1） R/I satisfies the general comparability; （2） R satisfies the general I-comparability condition; （3） The natural map B（R） → B（R/I） is surjective.     

Cogroups in algebras over an operad are free algebras

Let be an operad defined over a field of characteristic zero. Let R be a cogroup in the category of complete -algebras. In this article, we show that R is necessarily the completion of a free -algebra. We also handle the case of cogroups in connected graded algebras over an operad, and the case of groups in connected graded coalgebras over an operad. Received: August 26, 1996 and final version, February 4, 1998     

Commutative Non-Noetherian Rings with the Diamond Property

A ring R is said to have property (◇) if the injective hull of every simple R-module is locally Artinian. By landmark results of Matlis and Vamos, every commutative Noetherian ring has (◇). We give a systematic study of commutative rings with (◇), We give several general characterizations in terms of co-finite topologies on R and completions of R. We show that they have many properties of Noetherian rings, such as Krull intersection property, and recover several classical results of commutative Noetherian algebra, including some of Matlis and Vamos. Moreover, we show that a complete rings has (◇) if and only if it is Noetherian. We also give a few results relating the (◇) property of a local ring with that of its associated graded rings, and construct a series of examples.

     

Gorenstein Modules over Zariski Filtered Rings

Abstract

We study Gorenstein injective and projective modules over Zariski filtered rings and obtain relations between the Gorenstein dimensions on the category of filtered modules from the associated category of graded modules over the associated graded ring.     

Derived Equivalences of Upper Triangular Differential Graded Algebras

This paper generalises a result for upper triangular matrix rings to the situation of upper triangular matrix differential graded algebras. An upper triangular matrix DGA has the form (R, S, M) where R and S are differential graded algebras and M is a DG-left-R-right-S-bimodule. We show that under certain conditions on the DG-module M and with the existance of a DG-R-module X, from which we can build the derived category D(R), that there exists a derived equivalence between the upper triangular matrix DGAs (R, S, M) and (S, M′, R′), where the DG-bimodule M′ is obtained from M and X and R′ is the endomorphism differential graded algebra of a K-projective resolution of X.     

Relative Regular Modules. Applications to von Neumann Regular Rings

We use the concept of a regular object with respect to another object in an arbitrary category, in order to obtain the transfer of regularity in the sense of Zelmanowitz between the categories R −mod and S −mod, when S is an excellent extension of the ring R. Consequently, if S is an excellent extension of the ring R, then S is von Neumann regular ring if and only if R is also von Neumann regular ring. In the second part, using relative regular modules, we give a new proof of a classical result: the von Neumann regular property of a ring is Morita invariant.     

On the Buchsbaumness of the Associated Graded Ring of a One-Dimensional Local Ring

Let (R, m) be a Noetherian, one-dimensional, local ring, with |R/m|=∞. We study when its associated graded ring G(m) is Buchsbaum; in particular, we give a theoretical characterization for G(m) to be Buchsbaum not Cohen–Macaulay. Finally, we consider the particular case of R being the semigroup ring associated to a numerical semigroup S: we introduce some invariants of S, and we use them in order to give a necessary and a sufficient condition for G(m) to be Buchsbaum.     

<Emphasis Type="Italic">K</Emphasis><Subscript>0</Subscript> and rings over which the class of finitely generated strongly gorenstein projective modules is closed under extensions

In this paper, we consider the rings over which the class of finitely generated strongly Gorenstein projective modules is closed under extensions (called fs-closed rings). We give a characterization about the Grothendieck groups of the category of the finitely generated strongly Gorenstein projective R-modules and the category of the finitely generated R-modules with finite strongly Gorenstein projective dimensions for any left Noetherian fs-closed ring R.     

Weakly regular modules over normal rings

Under study are some conditions for the weakly regular modules to be closed under direct sums and the rings over which all modules are weakly regular. For an arbitrary right R-module M, we prove that every module in the category σ(M) is weakly regular if and only if each module in σ(M) is either semisimple or contains a nonzero M-injective submodule. We describe the normal rings over which all modules are weakly regular.     

Jacobson rings and rings strongly graded by an abelian group

LetR be ring strongly graded by an abelian groupG of finite torsion-free rank. Lete be the identity ofG, andR _e the component of degreee ofR. AssumeR _e is a Jacobson ring. We prove that graded subrings ofR are again Jacobson rings if eitherR _e is a left Noetherian ring orR is a group ring. In particular we generalise Goldie and Michlers’s result on Jacobson polycyclic group rings, and Gilmer’s result on Jacobson commutative semigroup rings of finite torsion-free rank.     

Free Pairs of Bicyclic Units in the Integral Group Ring of the Dihedral Group

Hirano studied the quasi-Armendariz property of rings, and then this concept was generalized by some authors, defining quasi-Armendariz property for skew polynomial rings and monoid rings. In this article, we consider unified approach to the quasi-Armendariz property of skew power series rings and skew polynomial rings by considering the quasi-Armendariz condition in mixed extension ring [R; I][x; σ], introducing the concept of so-called (σ, I)-quasi Armendariz ring, where R is an associative ring equipped with an endomorphism σ and I is an σ-stable ideal of R. We study the ring-theoretical properties of (σ, I)-quasi Armendariz rings, and we obtain various necessary or sufficient conditions for a ring to be (σ, I)-quasi Armendariz. Constructing various examples, we classify how the (σ, I)-quasi Armendariz property behaves under various ring extensions. Furthermore, we show that a number of interesting properties of an (σ, I)-quasi Armendariz ring R such as reflexive and quasi-Baer property transfer to its mixed extension ring and vice versa. In this way, we extend the well-known results about quasi-Armendariz property in ordinary polynomial rings and skew polynomial rings for this class of mixed extensions. We pay also a particular attention to quasi-Gaussian rings.     

Homological dimensions and semidualizing complexes

For finite modules over a local ring and complexes with finitely generated homology, we consider several homological invariants sharing some basic properties with projective dimension. In the second section, we introduce the notion of a semidualizing complex, which is a generalization of both a dualizing complex and a suitable module. Our goal is to establish some common properties of such complexes and the homological dimension with respect to them. Basic properties are investigated in Sec. 2.1. In Sec. 2.2, we study the structure of the set of semidualizing complexes over a local ring, which is closely related to the conjecture of Avramov-Foxby on the transitivity of the G-dimension. In particular, we prove that, for a pair of semidualizing complexes X ₁ and X ₂ such that G _X2, we have X ₂ ≃ X ₁ ⊗ _R ^L RHom _R (X ₁, X ₂). Specializing to the case of semidualizing modules over Artinian rings, we obtain a number of quantitative results for the rings possessing a configuration of semidualizing modules of special form. For the rings with m ³=0, this condition reduces to the existence of a nontrivial semidualizing module, and we prove a number of structural results in this case. In the third section, we consider the class of modules that contains the modules of finite CI-dimension and enjoys some nice additional properties, in particular, good behavior in short exact sequences. In the fourth section, we introduce a new homological invariant, CM-dimension, which provides a characterization for Cohen-Macaulay rings in precisely the same way as projective dimension does for regular rings, CI-dimension for locally complete intersections, and G-dimension for Gorenstein rings. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 30, Algebra, 2005.     

When Cyclic Modules Have Σ-Injective Hulls

Abstract

A theorem of Cartan-Eilenberg (Cartan, H., Eilenberg, S. (1956). Homological Algebra. Princeton: Princeton University Press, pp. 390.) states that a ring Ris right Noetherian iff every injective right module is Σ-incentive. The purpose of this paper is to study rings with the property, called right CSI, that, all cyclic right R-modules have Σ-injective hulls, i.e., injective hulls of cyclic right R-modules are Σ-injective. In this case, all finitely generated right R-modules have Σ-injective hulls, and this implies that Ris right Noetherian for a lengthy list of rings, most notably, for Rcommutative, or when Rhas at most finitely many simple right R-modules, e.g., when Ris semilocal. Whether all right CSIrings are Noetherian is an open question. However, if in addition, R/rad Ris either right Kasch or von Neuman regular (=VNR), or if all countably generated (sermisimple) right R-modules have Σ-injective hulls then the answer is affirmative. (See Theorem A.) We also prove the dual theorems for Δ-injective modules.