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.

<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>

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...

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...

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.

The main aim of the paper is to give a socle theory for Leavitt path algebras of arbitrary graphs. We use both the desingularization process and combinatorial methods to study Morita invariant properties concerning the socle and to characterize it, respectively. Leavitt path algebras with nonzero socle are described as those which have line points, and it is shown that the line points generate ...

First, we construct a bijection between the set of h-vectors and the set of socle-vectors of artinian algebras. As a corollary, we find the minimum codimension that an artinian algebra with a given socle-vector can have. Then, we study the main problem in the paper: determining when there is a unique socle-vector for a given h-vector. We solve the problem completely if the codimension is at mos...

SOCLE is a hybrid representation system in which cells of constraints are identified with slots of frame networks. Constraint formulas are maintained with respect to slots in frame networks and in turn provide for the dependency regulation of values on the frames. This paper illustrates the use of SOCLE and outlines the control structure decisions made for its design and implementation.

Zariski groups are @0-stable groups with an axiomatically given Zariski topology and thus abstract generalizations of algebraic groups. A large part of algebraic geometry can be developed for Zariski groups. As a main result, any simple smooth Zariski group interprets an algebraically closed eld, hence is almost an algebraic group over an algebraically closed eld.