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.

A module $M$ is called a $C_{21}$-module if, whenever $A$ and $B$ are submodules of with $A \cong B$, nonsingular direct summand $M$, then $M$. Various examples $C_{21}$-modules presented. Some basic properties these modules investigated. It shown that the class rings $R$ over which every $C_2$-module exactly right SI-rings. Also, we prove for ring $R$, $R$-module has $(C_{21})$ if only t-semis...

Double criticality and its nonlinear implications are considered for stratified N−layer shallow water flows with N = 1, 2, 3. Double criticality arises when the linearization of the steady problem about a uniform flow has a double zero eigenvalue. We find that there are two types of double criticality: non-semisimple (one eigenvector, and one generalized eigenvector), and semi-simple (two indep...

Let G be a semisimple, simply-connected algebraic group over an algebraically closed field of characteristic p > 0. We observe that the tensor product of the Steinberg module with a minuscule module is always indecomposable tilting. Although quite easy to prove, this fact does not seem to have been observed before. It has the following consequence: If p > 2h − 2 and a given tilting module has h...

Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle em...

Let L be a free Lie algebra of finite rank r over a field . For each positive integer n, denote the degree n homogeneous component of L by L. The group of graded algebra automorphisms of L may be identified with GLr; ‘ in such a way that L1 becomes the natural module, and then the L are referred to as the Lie powers of this module. Understanding the GLr; ‘-module structure of the L may be tho...

‎Let $S$ be an inverse semigroup with the set of idempotents $E$‎. We prove that the semigroup algebra $ell^{1}(S)$ is always‎ ‎$2n$-weakly module amenable as an $ell^{1}(E)$-module‎, ‎for any‎ ‎$nin mathbb{N}$‎, ‎where $E$ acts on $S$ trivially from the left‎ ‎and by multiplication from the right‎. ‎Our proof is based on a common fixed point property for semigroups‎.  

in this paper we will generalize  some of known results on the tight closure of an ideal to the tight closure of an ideal relative to a module .

A right R-module M is called a generalized q.f.d. module if every M-singular quotient has finitely generated socle. In this note we give several characterizations to this class of modules by means of weak injectivity, tightness, and weak tightness that generalizes the results in [25], Theorem 3. In particular, it is shown that a module M is g.q.f.d. iff every direct sum of M -singular M -inject...