Abstract We provide a proof of Jacobson’s theorem on derivations primitive rings with nonzero socle. Both and its formulation (in terms the so-called differential operators left vector spaces over division ring) underlie our paper. apply to describe standard operator real, complex, or quaternionic normed spaces. Indeed, when space is infinite-dimensional, every derivation such ring form $$A\rig...

Repeated root Cyclic and Negacyclic codes over Galois rings have been studied much less than their simple root counterparts. This situation is beginning to change. For example, repeated root codes of length p, where p is the characteristic of the alphabet ring, have been studied under some additional hypotheses. In each one of those cases, the ambient space for the codes has turned out to be a ...

A relatively compressed algebra with given socle degrees is an Artinian quotient A of a given graded algebra R/c, whose Hilbert function is maximal among such quotients with the given socle degrees. For us c is usually a “general” complete intersection and we usually require that A be level. The precise value of the Hilbert function of a relatively compressed algebra is open, and we show that f...

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.

We describe the tangent space to the parameter variety of all artin level quotients of a polynomial ring in n variables having specified socle degree and type. When n = 2, we relate this variety to the family of secants of the rational normal curve. With additional numerical hypotheses, we prove a projective normality theorem for the parameter variety in its natural Plücker embedding. AMS subje...

We first describe a situation in which every graded Betti number in the tail of the resolution of R J may be read from the socle degrees of R J . Then we apply the above result to the ideals J and J [q]; and thereby describe a situation in which the graded Betti numbers in the tail of the resolution of R/J [q] are equal to the graded Betti numbers in the tail of a shift of the resolution of R/J...

<abstract><p>Let $ R be a G graded commutative ring and M $-graded $-module. The set of all second submodules is denoted by Spec_G^s(M), it called the spectrum $. We discuss rings with Noetherian prime spectrum. In addition, we introduce notion Zariski socle explore their properties. also investigate Spec^s_G(M) topology from viewpoint being space.</p></abstract>

First we construct an interesting bijection between the set of h-vectors and the set of socle-vectors of artinian algebras. As a simple consequence, we find the minimum codimension that an artinian algebra with a given socle-vector can have. Then we address the main problem of this paper: determining when there is a unique socle-vector associated to a given h-vector. In Sections 3 and 5 we work...

Let R be a polynomial ring in r variables over an infinite field K, and denote by D a corresponding dual ring, upon which R acts as differential operators. We study type two graded level Artinian algebras A = R/I , having socle degree j. For each such algebra A, we consider the family of Artinian Gorenstein [AG] quotients of A having the same socle degree. By Macaulay duality, A corresponds to ...

In this document I describe how the standard way the socle operator is set up for a module, and a neater way is can be set up using the lattice of submodules. I indicate why this way is neater by showing how the construction can be iterated. I go through the various constructions first for the straight socle, and then for the socle relative to a given hereditary torsion theory.