Completions of Leavitt path algebras

We introduce a class of topologies on the Leavitt path algebra $L(\Gamma)$ of a finite directed graph and decompose a graded completion $\widehat{L}(\Gamma)$ as a direct sum of minimal ideals.Comment: 16 pages and 2 figure

Decomposition of Singular Matrices into Idempotents

In this paper we provide concrete constructions of idempotents to represent typical singular matrices over a given ring as a product of idempotents and apply these factorizations for proving our main results. We generalize works due to Laffey (Products of idempotent matrices. Linear Multilinear A. 1983) and Rao (Products of idempotent matrices. Linear Algebra Appl. 2009) to noncommutative setting and fill in the gaps in the original proof of Rao's main theorems. We also consider singular matrices over B\'ezout domains as to when such a matrix is a product of idempotent matrices.Comment: 15 page

On self-dual double circulant codes

Self-dual double circulant codes of odd dimension are shown to be dihedral in even characteristic and consta-dihedral in odd characteristic. Exact counting formulae are derived for them and used to show they contain families of codes with relative distance satisfying a modified Gilbert-Varshamov bound.Comment: 8 page

ADS modules

We study the class of ADS rings and modules introduced by Fuchs. We give some connections between this notion and classical notions such as injectivity and quasi-continuity. A simple ring R such that R is ADS as a right R-module must be either right self-injective or indecomposable as a right R-module. Under certain conditions we can construct a unique ADS hull up to isomorphism. We introduce the concept of completely ADS modules and characterize completely ADS semiperfect right modules as direct sum of semisimple and local modules.Comment: 7 page