Let $R$ be a commutative ring and let $M$ be an $R$-module. We define the small intersection graph $G(M)$ of $M$ with all non-small proper submodules of $M$ as vertices and two distinct vertices $N, K$ are adjacent if and only if $Ncap K$ is a non-small submodule of $M$. In this article, we investigate the interplay between the graph-theoretic properties of $G(M)$ and algebraic properties of $M...

For a graph G=(V,E) the edge ring k[G] is k[x1,…,xn]/I(G), where n=|V| and I(G) generated by {xixj;{i,j}∈E}. The conjecture we treat following.Conjecture 1. If has 2-linear resolution, then projective dimension of K[G], pd (k[G]), equals maximal degree vertex in G.As far as know, this first mentioned paper Gitler Valencia [7, Conjecture 4.13], there it called Eliahou-Villarreal conjecture. trea...

The Latino population in the United States continues to increase, diversify, and disperse geographically in dramatic fashion. Latinos are now the largest race/ethnic minority group in the United States, a fact not fully recognized by scholars, public officials, health practitioners, or people in the United States (Grieco & Cassidy, 2001). They are projected to constitute nearly one fourth of th...

A partially ordered group G = (G, +, ≤) is both a (not necessarily Abelian) group (G, +) (with binary operation + and identity element 0, where the inverse of a member a of G is denoted by −a) and a partially ordered set (G, ≤) in which a ≥ b implies that a+ c ≥ b+ c and c+ a ≥ c+ b for a, b, and c in G. An element a of G is positive if a ≥ 0. A partially ordered group (G, +, ≤) is called a lat...

Let G be a group which is the semidirect product of a normal subgroup N and some subgroup T . Let In(G), n ≥ 1, denote the powers of the augmentation ideal I(G) of the group ring Z(G). Using homological methods the groups Qn(G,H) = In−1(G)I(H)/In(G)I(H), H = G,N, T , are functorially expressed in terms of enveloping algebras of certain Lie rings associated with N and T , in the following cases:...

Ring chromosomes are rare chromosomal disorders that usually appear to occur de novo. A ring chromosome forms when due to deletion both ends of chromosome fuse with each other. Depending on the amount of chromosomal deletion, the clinical manifestations may be different. So, ring 18 syndrome is characterized by severe mental growth retardation as well as microcephaly, brain and ocular malformat...

Let G be a group with identity e. Let R be a G-graded commutative ring and let M be a graded R-module. The graded classical prime spectrum Cl.Specg(M) is defined to be the set of all graded classical prime submodule of M. The Zariski topology on Cl.Specg(M); denoted by ϱg. In this paper we establish necessary and sufficient conditions for Cl.Specg(M) with the Zariski topology to be a Noetherian...

Suppose that G is a linearly reductive group. Good degree bounds for generators of invariant rings were given in [2]. Here we study the minimal free resolution of the invariant ring. Recently it was shown that if G is a finite linearly reductive group, then the ring of invariants is generated in degree ≤ |G| (see [5, 6, 3]). This extends the classical result of Noether who proved the bound in c...

Let G be a connected semi-simple Lie group with torsion-free fundamental group. We show that the twisted equivariant KK-theory $$KK_{\bullet }^{G}(G/K, \tau _G^G)$$ of has ring structure induced from renowned K-theory $$K^{\bullet }_{K}(K, _K^K)$$ maximal compact subgroup K. give geometric description representatives in terms equivalence classes certain correspondences and obtain an optimal set...