let $r$ be a commutative ring with identity and $m$ be a finitely generated unital $r$-module. in this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. moreover, we investigate some conditions which imply that the module $m$ is the direct sum of some cyclic modules and free modules. then some properties of fitting ideals of...

Let G be an affine algebraic group over an algebraically closed field k and let H be a closed subgroup of G. If V is a rational H-module (a comodule for the coordinate ring of H) there is a now well-known notion of an induced module VI G for G, defined as the space Morph/~(G, V) of all H-equivariant morphisms from G to a finite dimensional subspace of V, with obvious G-action. The question aris...

We previously proposed evolutionary fuzzy systems of playing Ms.PacMan for the competitions. As a consequence of the evolution, reflective action rules such that PacMan tries to eat pills effectively until ghosts come close to PacMan are acquired. Such rules works well. However, sometimes it is too reflective so that PacMan go toward ghosts by herself in longer corridors. In this paper, a criti...

a module $m$ is called $emph{h}$-cofinitely supplemented if for every cofinite submodule $e$ (i.e. $m/e$ is finitely generated) of $m$ there exists a direct summand $d$ of $m$ such that $m = e + x$ holds if and only if $m = d + x$, for every submodule $x$ of $m$. in this paper we study factors, direct summands and direct sums of $emph{h}$-cofinitely supplemented modules. let $m$ be an $emph{h}$...

In this paper we defined the concept of module amenability of Banach algebras and module connes amenability of module dual Banach algebras.Also we assert the concept of module Arens regularity that is different with [1] and investigate the relation between module amenability of Banach algebras and connes module amenability of module second dual Banach algebras.In the following we studythe...

Let $\mathfrak{g}_{0}$ be a simple Lie algebra of type ADE and let $U'_{q}(\mathfrak{g})$ the corresponding untwisted quantum affine algebra. We show that there exists an action braid group $B(\mathfrak{g}_{0})$ on Grothendieck ring $\mathcal{K}_{t}(\mathfrak{g})$ Hernandez-Leclerc’s category $\mathcal{C}_{\mathfrak{g}}^{0}$. Focused case $A_{N-1}$, we construct family monoidal autofunctors $\{...

A W-algebra action is constructed via Hecke transformations on the equivariant Borel–Moore homology of Hilbert scheme points a nonreduced plane in three-dimensional affine space. The resulting W-module then identified to vacuum module. construction based generalization ADHM as well W-action moduli space instantons by Schiffmann and Vasserot.