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.

let s be a locally compact foundation semigroup with identity and                          be its semigroup algebra. let x be a weak*-closed left translation invariant subspace of    in this paper, we prove that  x  is invariantly  complemented in   if and  only if  the left ideal  of    has a bounded approximate identity. we also prove that a foundation semigroup with identity s is left amenab...

In this paper, two outwardly different graphs, namely, the zero-divisor graph [Formula: see text] and comaximal of ring all real-valued continuous functions having countable range, defined on any zero-dimensional space text], are investigated. It is observed that these graphs exhibit resemblance, so far as diameters, girths, connectedness, triangulatedness or hypertriangulatedness concerned. Ho...

Sentences of Visual Languages (VLs) may often be regarded as assemblies of pictorial objects with spatial relationships like “above” or “contains” between them, i.e. their representations are a kind of directed graphs. Such a spatial relationship graph is often complemented by a more abstract graph, which provides information about the syntax (and the semantics) of the visual sentence in a more...

Let $L$ be a finite-dimensional Lie algebra. We say a subalgebra $H$ of $L$ is permutably complemented in $L$ if there is a subalgebra $K$ of $L$ such that $L=H+K$ and $Hcap K=0$. Also, if every subalgebra of $L$ is permutably complemented in $L$, then $L$ is called completely factorisable. In this article, we consider the influence of these concepts on the structure of a Lie algebra, in partic...

Rosenthal [R] introduced the Lp space Xp in 1971. Among its interesting properties are that it contains and is contained in isomorphs of lp ⊕ l2, but is not isomorphic to a complemented subspace of lp⊕ l2. These properties have made Xp rather resistant to standard approaches to classifying its complemented subspaces. For example it was first proved that Xp was primary in [JO2] where the device ...