A Note on the Commutativity of Topologizing Filters on Noetherian Rings

Let R be a ring with identity, and R-ill denote the set of all left topologizing filters on R. In this paper, we give a sufficient condition for the commutativity of R-ill under the hypothesis of left Noetherianness, as well as some examples.

Noetherian Skew Group Rings of F.C Groups

Let R ＊θ G be the skew group ring with a F.C group G and the group homomrphism 8 from G to Aut（R）, the group of automorphisms of the ring R. In this paper,the necessary and sufficient condition such that R ＊θ G will be Noetherian is given, which generalizes the results of LG. connel.

EXPLICIT DESCRIPTION OF A CLASS OF INDECOMPOSABLE INJECTIVE MODULES

Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module R/p.

Maximal codes

In this work we try to introduce the concept of Maximal codes that are built over rings, more precisely we will give Maximal codes for special rings, Namely that the notion of maximal codes has been used by Chritophe Chapote, these maximal codes are constructed over finite fields, and these codes are used for coding and decoding.

Gorenstein projective modules over rings of Morita contexts

Under semi-weak and weak compatibility conditions of bimodules,we establish necessary and sufficient conditions of Gorenstein-projective modules over rings of Morita contexts with one bimodule homomorphism zero.This extends greatly the results on triangular matrix Artin algebras and on Artin algebras of Morita contexts with two bimodule homomorphisms zero in the literature,where only sufficient conditions are given under a strong assumption of compatibility of bimodules.An application is provided to describe Gorenstein-projective modules over noncommutative tensor products arising from Morita contexts.Our results are proved under a general setting of noetherian rings and modules instead of Artin algebras and modules.

The Classification of the Annihilating-Ideal Graphs of Commutative Rings

Let R be a commutative ring and A（R） be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG（R） with the vertex set A（R）＊ = A（R）／{（0）} and two distinct vertices I and J are adjacent if and only if IJ = （0）. Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and w（AG（R）） = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.

Envelopes of Commutative Rings

Given a significative class F of commutative rings, we study the precise conditions under which a commutative ring R has an S-envelope. A full answer is obtained when F is the class of fields, semisimple commutative rings or integral domains. When .F is the class of Noetherian rings, we give a full answer when the Krull dimension of R is zero and when the envelope is required to be epimorphic. The general problem is reduced to identifying the class of non-Noetherian rings having a monomorphic Noetherian envelope, which we conjecture is the empty class.

Injective Dimension of Weyl Algebras over Affine Coefficient Rings

The concern of this paper is to derive formulas for the injective dimension of the n-th Weyl algebra A_n(R)in case k is a field of characteristic zero and R is a commutative affine k-algebra of finite injective dimension.For the case n=1 we prove a more general result from which the above result follows.Such formulas can be viewed as generalizations of the corresponding results given by J.C.McConnell in the case R has finite global dimension.

Characterization of Weak I Sequences

In [5], Zhou defined the notion of weak I sequences and characterized such se-quences by Koszul cohomology and local cohomology methods. The aim of this paper is to characterize weak I sequences by means of Ext functor.

TOWARD A THEORY OF WEAK I SEQUENCES

The author introduces a notion of weakIsequences and characterizes such sequences by means of homological methods.This notion extends the notion of weakMsequences and thus extends the notions of generalized Cohen Macaulay modules and Buchsbaum modules to more general cases.

Homological behavior of idempotent subalgebras and Ext algebras

Let A be a(left and right) Noetherian ring that is semiperfect. Let e be an idempotent of A and consider the ring Γ :=(1-e)A(1-e) and the semi-simple right A-module Se := e A/e rad A. In this paper, we investigate the relationship between the global dimensions of A and Γ, by using the homological properties of Se. More precisely, we consider the Yoneda ring Y(e) := Ext_A~*(Se, Se) of e. We prove that if Y(e) is Artinian of finite global dimension, then A has finite global dimension if and only if so does Γ. We also investigate the situation where both A and Γ have finite global dimension. When A is Koszul and finite dimensional, this implies that Y(e) has finite global dimension. We end the paper with a reduction technique to compute the Cartan determinant of Artin algebras. We prove that if Y(e) has finite global dimension, then the Cartan determinants of A and Γ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture.

On Standard Ideals

In the paper, we study a class of standard ideals which are more general than the m-primary standard ideals discussed in [2]. We will prove an important equality concerning I-weak sequences; thus a generalization of the equality of [2] is established.