In this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $Z_{2}$. We also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.

Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera-JiaReiner quasi-symmetric function, and the invariant G introduced by the first author, are valuative. In this paper we construct the Z-modules of all Z-valued valuative functions for labeled matroids and polymatroids on a fixed ground set, and their unlabeled counterparts, the Z-modules of valuat...

Let H be a quasi-Hopf algebra. We show that any H-bimodule coalgebra C for which there exists an H-bimodule coalgebra morphism ν : C → H is isomorphic to what we will call a smash product coalgebra. To this end, we use an explicit monoidal equivalence between the category of two-sided two-cosided Hopf modules over H and the category of left Yetter-Drinfeld modules over H. This categorical metho...

Given a quasi-hereditary superalgebra A, the first author and R. Muth have defined generalized Schur bi-superalgebras TA(n) proved that these algebras are again quasi-hereditary. In particular, comes with family of standard modules. Developing work Donkin Mathieu on good filtrations, we prove tensor product modules over has filtration.

In this paper, we present a new approach for mining weighted networks to identify densely connected modules such as quasi-cliques. Quasi-cliques are densely connected subnetworks in a network. Detecting quasi-cliques is an important topic in data mining, with applications such as social network study and biomedicine. Our approach has two major improvements upon previous work. The first is the u...

let $r$ be a commutative noetherian ring with non-zero identity, $fa$ an ideal of $r$, and $x$ an $r$--module. here, for fixed integers $s, t$ and a finite $fa$--torsion $r$--module $n$, we first study the membership of $ext^{s+t}_{r}(n, x)$ and $ext^{s}_{r}(n, h^{t}_{fa}(x))$ in the serre subcategories of the category of $r$--modules. then, we present some conditions which ensure the exi...

we introduce the notions of t-dual rickart and strongly t-dual rickart modules. we provide several characterizations and investigate properties of each of these concepts. it is shown that every free (resp. finitely generated free) $r$-module is t-dual rickart if and only if $overline{z}^2(r)$ is a  direct summand of $r$ and end$(overline{z}^2(r))$ is a semisimple (resp. regular) ring. it is sho...