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 $m$...

It is shown that a simple vertex operator algebra V is rational if and only if its Zhu algebra A(V ) is semisimple and each irreducible admissible V -module is ordinary. A contravariant form on a Verma type admissible V -module is constructed and the radical is exactly the maximal proper submodule. As an application the rationality of V + L for any positive definite even lattice is obtained.

We consider the problem of decomposing tensor powers of the fundamental level 1 highest weight representation V of the affine Kac-Moody algebra g(E9). We describe an elementary algorithm for determining the decomposition of the submodule of V ⊗n whose irreducible direct summands have highest weights which are maximal with respect to the null-root. This decomposition is based on Littelmann’s pat...

We consider the problem of decomposing tensor powers of the fundamental level 1 highest weight representation V of the affine Kac-Moody algebra g(E9). We describe an elementary algorithm for determining the decomposition of the submodule of V ⊗n whose irreducible direct summands have highest weights which are maximal with respect to the null-root. This decomposition is based on Littelmann’s pat...

We consider the problem of decomposing tensor powers of the fundamental level 1 highest weight representation V of the affine Kac-Moody algebra g(E9). We describe an elementary algorithm for determining the decomposition of the submodule of V ⊗n whose irreducible direct summands have highest weights which are maximal with respect to the null-root. This decomposition is based on Littelmann’s pat...

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 ...

The blob algebra is a finite-dimensional quotient of the Hecke type B which almost always quasi-hereditary. We construct indecomposable tilting modules for over field characteristic 0 in doubly critical case. Every module maximal highest weight either projective or an extension simple by module. Moreover, every submodule weight. conclude that graded Weyl filtration multiplicities this case are ...

A finitely generated $R$-module is said to be a module of type ($F_r$) if its $(r-1)$-th Fitting ideal is the zero ideal and its $r$-th Fitting ideal is a regular ideal. Let $R$ be a commutative ring and $N$ be a submodule of  $R^n$ which is generated by columns of  a matrix $A=(a_{ij})$ with $a_{ij}in R$ for all $1leq ileq n$, $jin Lambda$, where $Lambda $ is a (possibly infinite) index set.  ...

The modular multilevel converter (MMC) is receiving extensive research interests in high/medium voltage applications due to its modularity, scalability, reliability, high voltage capability and excellent harmonic performance. The submodule capacitors are usually quite bulky since they have to withstand fundamental frequency voltage fluctuations. To reduce the capacitance of these capacitors, th...

12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x1, . . . , xn is any maximal set of linearly independent elements of M . Prove that N is isomorphic to Rn and that the quotient M/N is a torsion Rmodule. (b) Prove conversely that if M contains a submodule N that is free of rank n (i.e., N › Rn) such that the quotient M/N is a torsion R-module then M ha...