Abstract We introduce and study model-theoretic connected components of rings as an analogue definable groups. develop their basic theory use them to describe both the classical Bohr compactifications rings. then explicitly calculate some matrix groups, such discrete Heisenberg group ${\mathrm {UT}}_3({\mathbb {Z}})$ , continuous {R}})$ and, more generally, groups upper unitriangular invertible...

let $d$ be a digraph with skew-adjacency matrix $s(d)$‎. ‎the skew‎ ‎energy of $d$ is defined as the sum of the norms of all‎ ‎eigenvalues of $s(d)$‎. ‎two digraphs are said to be skew‎ ‎equienergetic if their skew energies are equal‎. ‎we establish an‎ ‎expression for the characteristic polynomial of the skew‎ ‎adjacency matrix of the join of two digraphs‎, ‎and for the‎ ‎respective skew energ...

We study certain Schur functors which preserve singularity categories of rings and we apply them to study the singularity category of triangular matrix rings. In particular, combining these results with Buchweitz–Happel’s theorem, we can describe singularity categories of certain non-Gorenstein rings via the stable category of maximal Cohen–Macaulay modules. Three concrete examples of finitedim...