The concept of a combinatorial decomposition of a graded K algebra was introduced by Baclawski-Garsia [4], and they showed that every (finitelygenerated) graded K algebra has such a decomposition. The purpose of this paper is to prove some general properties of combinatorial decompositions, which are useful for finding such decompositions. We then show how to compute combinatorial decomposition...

The vacuolar ATPase (V-ATPase) proton pump sustains cellular pH homeostasis, and its inhibition triggers numerous stress responses. However, the cellular mechanisms involved remain largely elusive in cancer cells. We studied V-ATPase in the prostate cancer (PCa) cell line PC-3, which has characteristics of highly metastatic PCa. V-ATPase inhibitors impaired endo-lysosomal pH, vesicle traffickin...

In this paper, we study non-central almost subnormal subgroups of the multiplicative group a division ring satisfying nonzero generalized rational identity. The main result generalizes Chiba’s theorem on subgroups. As an application, get algebraic last has several corollaries which generalize completely or partially some previous results.

An R-module is called semi-endosimple if it has no proper fully invariant essential submodules. For a quasi-projective retractable module MR we show that M is finitely generated semi-endosimple if and only if the endomorphism ring of M is a finite direct sum of simple rings. For an arbitrary module M , conditions equivalent to the semi-endosimplicity of its quasi-injective hull are found. As co...

A graph G= (V, E) is called a hyper-ring with N nodes (N-HR for short) if V = {0, ..., N-1} and E = {(u,v)| v-u modulo N is a power of 2}. In other words, a hyper-ring with N nodes is a graph in which all of the edges connect nodes that are at distances of powers of 2 from each other. Hyper-rings are a multi-machine organization that is, in a sense, a generalization of the standard hyper-cube a...