Let k be a field of finite characteristic p, and G a finite group acting on the left on a finite dimensional k-vector space V . Then the dual vector space V ∗ is naturally a right kG-module, and the symmetric algebra of the dual, R := Sym(V ∗), is a polynomial ring over k on which G acts naturally by graded algebra automorphisms, and if k is algebraically closed can be regarded as the space k[V...

let $g=(v,e)$ be a simple graph. a set $ssubseteq v$ isindependent set of $g$,  if no two vertices of $s$ are adjacent.the  independence number $alpha(g)$ is the size of a maximumindependent set in the graph. in this paper we study and characterize the independent sets ofthe zero-divisor graph $gamma(r)$ and ideal-based zero-divisor graph $gamma_i(r)$of a commutative ring $r$.

In this paper, we introduce the notion of $G_\infty$-ring spectra. These are globally equivariant homotopy types with a structured multiplication, giving rise to power operations on their and cohomology groups. We illustrate structure by analysing when Moore spectrum can be endowed structure. Such $G_\infty$-structures correspond underlying ring, indexed Burnside ring. exhibit close relation be...

let $r$ be a commutative ring and let $m$ be an $r$-module. we define the small intersection graph $g(m)$ of $m$ with all non-small proper submodules of $m$ as vertices and two distinct vertices $n, k$ are adjacent if and only if $ncap k$ is a non-small submodule of $m$. in this article, we investigate the interplay between the graph-theoretic properties of $g(m)$ and algebraic properties of $m...