Introduction. Let A be a noetherian ring. When A is commutative (of finite Krull dimension), A is said to be Gorenstein if its injective dimension is finite. If A has finite global dimension, one says that A is regular. If A is arbitrary, these hypotheses are not sufficient to obtain similar results to those of the commutative case. To remedy this problem, M. Auslander has introduced a suppleme...

The Doob graph D(m,n) is a distance-regular graph with the same parameters as the Hamming graph H(2m+n, 4). The maximum independent sets in the Doob graphs are analogs of the distance-2 MDS codes in the Hamming graphs. We prove that the logarithm of the number of the maximum independent sets in D(m,n) grows as 2(1+o(1)). The main tool for the upper estimation is constructing an injective map fr...

A sharp base B is a base such that whenever (Bi)i<ω is an injective sequence from B with x ∈ i<ω Bi, then { ⋂ i<n Bi : n < ω} is a base at x. Alleche, Arhangel’skĭı and Calbrix asked: if X has a sharp base, must X × [0, 1] have a sharp base? Good, Knight and Mohamad claimed to construct an example of a Tychonoff space P with a sharp base such that P × [0, 1] does not have a sharp base. However,...

Let R be a ring and let M be a right R-module with S End MR . M is called almost general quasiprincipally injective or AGQP-injective for short if, for any 0/ s ∈ S, there exist a positive integer n and a left ideal Xsn of S such that s / 0 and lS Ker s Ss ⊕ Xsn . Some characterizations and properties of AGQP-injective modules are given, and some properties of AGQP-injective modules with additi...

We show, in full generality, that Lusztig’s a-function describes the projective dimension of both indecomposable tilting modules and indecomposable injective modules in the regular block of the BGG category O, proving a conjecture from the first paper. On the way we show that the images of simple modules under projective functors can be represented in the derived category by linear complexes of...

In this article we study injective representations of infinite quivers. We classify the indecomposable injective representations of trees and then describe Gorenstein injective and projective representations of barren trees.

We give some new properties of almost injective modules and their endomorphism rings, and also provide conditions as to when a direct sum of almost injective (or CS) modules is again almost injective (or CS) in some special cases..

We consider a symmetric monoidal closed category V = (V ,⊗, I, [−,−]) together with a regular injective object Q such that the functor [−, Q] : V → V op is comonadic and prove that in such a category, as in the monoidal category of abelian groups, a morphism of commutative monoids is an effective descent morphism for modules if and only if it is a pure monomorphism. Examples of this kind of mon...