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.

Introduction Let G be a simply connected and connected semisimple group over an algebraically closed field k of characteristic p > 0. T ⊂ G is a maximal torus and R is the root system relative to T . X(T ) is the weight lattice. Let B ⊃ T be a Borel subgroup corresponding to the negative roots R− = R. Denote by Gr the r-th Frobenius kernel of G. The socle and radical structures of the cohomolog...

Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle em...

In our paper referred to above we claim to enumerate all …nite primitive linear groups of prime degree r over C with a nonabelian socle. However, the case where the socle is imprimitive was overlooked. In the present paper we deal with this case to complete the classi…cation. 2000 Mathematics Subject Classi…cation: 20H20 20C15 20C33 In the paper referred to above, we state a theorem (Theorem 1....