For any ideal I in a Noetherian local ring or graded standard K-algebra over field K, we introduce the socle module Soc(I), whose components give us of powers I. It is observed that Soc(I) finitely generated fiber cone In case S polynomial and all I⊆S have linear resolution, define Soc∗(I), which Rees edge graph for classes polymatroidal ideals, study structure their modules.

let $r$ be a commutative ring and let $m$ be an $r$-module. in this article, we introduce the concept of the zariski socles of submodules of $m$ and investigate their properties. also we study modules with noetherian second spectrum and obtain some related results.

Let R = k[x 1 ,. .. , xn] and let I be the ideal of n + 1 generically chosen forms of degrees d 1 ≤ · · · ≤ d n+1. We give the precise graded Betti numbers of R/I in the following cases: • n = 3; • n = 4 and 5 i=1 d i is even; • n = 4, 5 i=1 d i is odd and d 2 + d 3 + d 4 < d 1 + d 5 + 4; • n is even and all generators have the same degree, a, which is even; • (n+1 i=1 d i) − n is even and d 2 ...

In a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its ...

We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function (1, 3, 6, 6, 3, 1), we give a complete answer in every characteristic by translating the problem to one of...

A certain class of rank two pointed Hopf algebras is considered. The simple modules of their Drinfel’d double is described using Radford’s method [Rad03]. The socle of the tensor product of two such modules is computed and a formula similar to the one in [Che00] is obtained in some conditions. Cases when such a tensor product is completely irreducible are also given in the last section.