2-polyhedra for which every homotopy domination over itself is a homotopy equivalence

Modules for which every non-cosingular submodule is a summand

‎A module $M$ is lifting if and only if $M$ is amply supplemented and‎ ‎every coclosed submodule of $M$ is a direct summand‎. ‎In this paper‎, ‎we are‎ ‎interested in a generalization of lifting modules by removing the condition‎"‎amply supplemented‎" ‎and just focus on modules such that every non-cosingular‎ ‎submodule of them is a summand‎. ‎We call these modules NS‎. ‎We investigate some gen...

Classification of graphs based on homotopy equivalence. Homotopy equivalent graphs. Basic graphs and complexity of homotopy equivalence classes of graphs

The main concept of this classification is that graphs belonging to the same equivalence class have similar topological properties. Graphs are called homotopy equivalent if one of them can be converted to the other one by contractible transformations. A basic representative and complexity of a homotopy equivalence class are defined and investigated. Finding a basic representative within a homot...

Homotopy equivalence of isospectral graphs

In previous work we defined a Quillen model structure, determined by cycles, on the category Gph of directed graphs. In this paper we give a complete description of the homotopy category of graphs associated to our model structure. We endow the categories of N-sets and Z-sets with related model structures, and show that their homotopy categories are Quillen equivalent to the homotopy category H...

Rings for which every simple module is almost injective

We introduce the class of “right almost V-rings” which is properly between the classes of right V-rings and right good rings. A ring R is called a right almost V-ring if every simple R-module is almost injective. It is proved that R is a right almost V-ring if and only if for every R-module M, any complement of every simple submodule of M is a direct summand. Moreover, R is a right almost V-rin...

Strong Homotopy Equivalence of 3-manifolds