Search

In this paper, the notions of (p, λ)-Koszul algebra and (p, λ)-Koszul module are introduced. Some criteria theorems for a positively graded algebra A to be (p, λ)-Koszul are given. The notion of weakly (p, λ)-Koszul module is defined as well and let WKλp(A) denote the category of weakly (p, λ)-Koszul modules. We show that M ∈ WKλp(A) if and only if it can be approximated by (p, λ)-Koszul submodules, which is equivalent to that G(M) is a (p, λ)-Koszul module, where G(M) denotes the associated graded module of M. As applications, the relationships of the minimal graded projective resolutions of M, G(M) and (p, λ)-Koszul submodules are established. In particular, for a module M ∈ WKλp(A) we prove that ⊕i≥0 ExtAi(M,A0) ∈ gr0(E(A)), we also get as a consequence that the finitistic dimension conjecture is valid in WKλp(A) under certain conditions.

A binary system in information granules is introduced in this article. Based on this system, a search engine of Web page abstract information is constructed, where the search results are processed by granular reasoning. Firstly, definitions for information granule and preprocess, such as accord degree, granular matrix, granular relation matrix, constrained relation matrix, association degree and so on, have been represented. Secondly, with these definitions, a personalized meta-info search engine experimental system is developed. Finally, the experimental results show that the search results returned by this system match userpsilas expectation effectively.

In this paper, the extension closure of the subcategory of mod R consisting of k-torsionfree modules with respect to a generalized tilting bimodule is discussed. Some classical results related to the extension closure of k-torsionfree modules are generalized and strengthened. Let RωS be a cotilting bimodule, the notion of left ⊥ω-approximation dimension is introduced, and as an application, we give a condition such that a ω-k-syzygy module to be ω-k-torsionfree.