Let $$Q \rightarrow R$$ be a surjective homomorphism of Noetherian rings such that Q is Gorenstein and R as Q-bimodule admits finite resolution by modules which are projective on both sides. We define an adjoint pair functors between the homotopy category totally acyclic R-complexes Q-complexes. This analogous to classical module categories Q. As consequence, we obtain precise notion approximat...

For any reduced commutative $f$-ring with identity and bounded inversion, we show that a condition which is obviously necessary for the socle of the ring to coincide with the socle of its bounded part, is actually also sufficient. The condition is that every minimal ideal of the ring consist entirely of bounded elements. It is not too stringent, and is satisfied, for instance, by rings of ...

A vertex in a graph totally dominates another vertex if they are adjacent. A sequence of vertices in a graph G is called a total dominating sequence if every vertex v in the sequence totally dominates at least one vertex that was not totally dominated by any vertex that precedes v in the sequence, and at the end all vertices of G are totally dominated. While the length of a shortest such sequen...

Abstract In this paper we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. More precisely, given a totally nonholonomic distribution of rank 2 manifold, investigate size set points that can be reached by singular horizontal paths starting from point and it has Hausdorff dimension at most 1. fact, provided lengths curves under consideration are ...