In this paper we introduce abstract string modules and give an explicit bijection between the submodule lattice of module perfect matching corresponding snake graph. particular, make direct correspondence a For every define Coxeter element in symmetric group. We then establish given by interval weak Bruhat order determined element. Using graphs, new concise formulation graph calculus.

Let M be an R-module and 0 6= f ∈ M∗ = Hom(M, R). The graph Γf (M) is a graph with vertices Z f (M) = {x ∈ M \ {0} | xf(y) = 0 or yf(x) = 0 for some non-zero y ∈ M}, in which non-zero elements x and y are adjacent provided that xf(y) = 0 or yf(x) = 0, which introduced and studied in [3]. In this paper we associate an undirected submodule based graph ΓfN (M) for each submodule N of M with vertic...

the submodules with the property of the title ( a submodule $n$ of an $r$-module $m$ is called strongly dense in $m$, denoted by $nleq_{sd}m$, if for any index set $i$, $prod _{i}nleq_{d}prod _{i}m$) are introduced and fully investigated. it is shown that for each submodule $n$ of $m$ there exists the smallest subset $d'subseteq m$ such that $n+d'$ is a strongly dense submodule of $m$...

We present the results of a series of computer simulations on the joint motion of two kinks in the elementary cellular automaton Rule 18. These show a remarkably close approximation to the case of independent annihilating random walks except at close range interaction where combinatorial conditions impose an asymmetry on the dynamics. Attempts to analyze the underlying state transition graph ar...

All rings are commutative with 1 6= 0, and all modules are unital. The purpose of this paper is to investigate the concept of 2-absorbing primary submodules generalizing 2-absorbing primary ideals of rings. Let M be an R-module. A proper submodule N of an R-module M is called a 2-absorbing primary submodule of M if whenever a, b ∈ R and m ∈M and abm ∈ N , then am ∈M -rad(N) or bm ∈M -rad(N) or ...

Corollary 1.2 Let G be an almost simple classical group with natural module V of dimension n. If G is linear, assume that G does not contain a graph automorphism. Let 2 ≤ k < n − 1, and let K be the stabilizer of a nondegenerate or totally singular k-space of V . Let P be the stabilizer of a singular 1-space of V . Then the permutation module 1P is a submodule of 1 G K unless one of the followi...