Let Λ be a quasi k-Gorenstein ring. For each dth syzygy module M in mod Λ (where 0 ≤ d ≤ k − 1), we obtain an exact sequence 0 → B → M ⊕ P → C → 0 in mod Λ with the properties that it is dual exact, P is projective, C is a (d + 1)st syzygy module, B is a dth syzygy of Ext Λ (D(M),Λ) and the right projective dimension of B ∗ is less than or equal to d − 1. We then give some applications of such ...

Evaluators and researchers often have to deal with situations in which conventional research tools are impossible to use, either because of the characteristics of a population or unclear research variables. This paper presents a technique that succeeds in overcoming this kind of problem--a projective technique, but one that differs from the usual approach to projective techniques. The approach ...

During recent years, several studies have revealed that human-dog relationships are based on a well-established and complex bond. There is now evidence suggesting that the dog-human affectional bond can be characterized as an "attachment". The present study investigated possible association between the owners' attachment profile assessed throughout a new semi-projective test (the 9 Attachment P...

We initiate the study of 1-torsion of finite modules over two-sided noetherian semiperfect rings. In particular, we give a criterion for determining when the 1-torsion submodule contains minimal generators of the module. We also provide an explicit construction for a projective cover of the submodule generated by the torsion elements in the top of the module. Some of the obtained results hold w...

Let T be a Noetherian ring and f a nonzerodivisor on T . We study concrete necessary and sufficient conditions for a module over R = T/(f) to be weakly liftable to T , in the sense of Auslander, Ding and Solberg. We focus on cyclic modules and get various positive and negative results on the lifting and weak lifting problems. For a module over T we define the loci for certain properties: liftab...

Let U denote the half quantum group for a fixed simple Lie algebra. We examine some properties and representation of U. We prove that the Hopf algebra U is not quasi-cocommutative, and hence the category of left U-module is not a braided monoidal category. In the weight module category W , we describe all the simple objects and the projective objects. We also describe all simple Yetter-Drinfel’...

Let R be a commutative Noetherian local ring with residue class field k. In this paper, we mainly investigate direct summands of the syzygy modules of k. We prove that R is regular if and only if some syzygy module of k has a semidualizing summand. After that, we consider whether R is Gorenstein if and only if some syzygy module of k has a G-projective summand.

We characterize the finiteness of the representation type of an artin algebra in terms of the behavior of the projective covers and the injective envelopes of the simple modules with respect to the infinite radical of the module category. In case the algebra is representation-finite, we show that the nilpotency of the radical of the module category is the maximal depth of the composites of thes...