Let V be a finite-dimensional F -vector space, and let T : V → V be a linear transformation. In this note we prove that V is a direct sum of cyclic T -invariant subspaces. More specifically, we prove that V is a direct sum of cyclic T -invariant subspaces whose annihilators are generated by powers of irreducible polynomials, and that the collection of these polynomials is uniquely determined. R...

The Garsia–Haiman module is a bigraded Sn-module whose Frobenius image Macdonald polynomial. method of orbit harmonics promotes an Sn-set X to graded polynomial ring. can be applied prove cyclic sieving phenomena which notion that encapsulates the fixed-point structure finite group action on set. By applying this idea module, we provide results regarding enumeration matrices are invariant under...

We consider a cyclic group of order p acting on a module incharacteristic p and show how to reduce the calculation of the symmetric algebra to that of the exterior algebra. Consider a cyclic group of order p acting on a polynomial ring S = k[x1, . . . , xr], where k is a field of characteristic p; this is equivalent to the symmetric algebra S∗(V ) on the module V generated by x1, . . . , xr. We...

Introduction 2 1. Preliminaries on conformal algebras and modules 4 2. Basic definitions 9 3. Extensions and deformations 13 4. Homology 17 5. Exterior multiplication, contraction, and module structure 18 6. Cohomology of conformal algebras and their annihilation Lie algebras 19 6.1. Cohomology of the basic complex 19 6.2. Cohomology of the reduced complex 21 6.3. Cohomology of conformal algebr...

In the mid-1960s Borevič and Faddeev initiated the study of the Galois module structure of groups of pth-power classes of cyclic extensions K/F of pth-power degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois modules for all base fields F . In 1947 Šafarevič initiated the study of Galois groups of maximal pextension...

We introduce an additive but not F4-linear map S from Fn4 to F 4 and exhibit some of its interesting structural properties. If C is a linear [n, k, d]4-code, then S(C) is an additive (2n, 2 , 2d)4-code. If C is an additive cyclic code then S(C) is an additive quasi-cyclic code of index 2. Moreover, if C is a module θ-cyclic code, a recently introduced type of code which will be explained below,...

The object of this paper is to study the relationship between certain projective modules and their endomorphism rings. Specifically, the basic problem is to describe the projective modules whose endomorphism rings are (von Neumann) regular, local semiperfect, or left perfect. Call a projective module regular if every cyclic submodule is a direct summand. Thus a ring is a regular module if it is...

In this article we present a new class of modules which is named as principally -lifting modules. This termed by Principally in work defined as, module called if for every cyclic submodule with , there decomposition such that and g-small . Thus, ring it -module. We determined structure. Several characterizations, properties, instances are described these modules'.

Let $R\ $be a commutative ring with $1\neq0$ and $M$ be an $R$-module. Suppose that $S\subseteq R\ $is multiplicatively closed set of $R.\ $Recently Sevim et al. in \cite{SenArTeKo} introduced the notion $S$-prime submodule which is generalization prime used them to characterize certain classes rings/modules such as submodules, simple modules, torsion free modules,\ $S$-Noetherian modules etc. ...