we first characterize $tau$-complemented modules with relative (pre)-covers. we also introduce an extending module relative to $tau$-pure submodules on a hereditary torsion theory $tau$ and give its relationship with $tau$-complemented modules.

We first characterize $tau$-complemented modules with relative (pre)-covers. We also introduce an extending module relative to $tau$-pure submodules on a hereditary torsion theory $tau$ and give its relationship with $tau$-complemented modules.

Relative to a hereditary torsion theory $tau$ we introduce a dimension for a module $M$, called {em $tau$-rank of} $M$, which coincides with the reduced rank of $M$ whenever $tau$ is the Goldie torsion theory. It is shown that the $tau$-rank of $M$ is measured by the length of certain decompositions of the $tau$-injective hull of $M$. Moreover, some relations between the $tau$-rank of $M$ and c...

relative to a hereditary torsion theory $tau$ we introduce a dimension for a module $m$, called {em $tau$-rank of} $m$, which coincides with the reduced rank of $m$ whenever $tau$ is the goldie torsion theory. it is shown that the $tau$-rank of $m$ is measured by the length of certain decompositions of the $tau$-injective hull of $m$. moreover, some relations between the $tau$-rank of $m$ and c...

Let $A$ be a Banach ternary algebra over a scalar field $Bbb R$ or $Bbb C$ and $X$ be a ternary Banach $A$--module. Let $sigma,tau$ and $xi$ be linear mappings on $A$, a linear mapping $D:(A,[~]_A)to (X,[~]_X)$ is called a Lie ternary $(sigma,tau,xi)$--derivation, if $$D([a,b,c])=[[D(a)bc]_X]_{(sigma,tau,xi)}-[[D(c)ba]_X]_{(sigma,tau,xi)}$$ for all $a,b,cin A$, where $[abc]_{(sigma,tau,xi)}=ata...

In this corrigendum, we give a correction of one result in reference [1].

Comparing the bounded derived categories of an algebra and endomorphism a given support {\tau}-tilting module, we find relation between dimensions module.

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n)) = 0 for n < 0, V(0)=C1, and the contragredient module V' is isomorphic to V as a V-module; (ii) every N-gradable weak V-module is completely reducible; (iii) V is C(2)-cofinite. We announce a proof of the Verlinde conjecture for V, that is, of the statement that the matrices formed by the fusion rules among...

Let G be a semisimple Lie group with associated symmetric space D, and let Γ ⊂ G be a cocompact arithmetic group. Let L be a lattice inside a ZΓ-module arising from a rational finite-dimensional complex representation of G. Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion subgroup Hi(Γk;L )tors as Γk ranges over a tower of congruence subgrou...

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin-Sternberg geometric quantization conjecture holds if M...