let $mathcal{x}$ be a class of $r$-modules‎. ‎in this paper‎, ‎we investigate ;$mathcal{x}$-injective (projective) and dg-$mathcal{x}$-injective (projective) complexes which are generalizations of injective (projective) and dg-injecti‎‎ve (projective) complexes‎. ‎we prove that some known results can be extended to the class of ;$mathcal{x}$-injective (projective) and dg-$mathcal{x}$-injective ...

In this paper, we investigate the notions of X ⊥ -projective, id="M2"> -injective, and id="M3"> -flat modules give some characterizations these modules, where id="M4"> is a class left modules. We prove that all id="M5"> -projective Kaplansky. Further, if id="M6"...

let r be a commutative noetherian ring. we study the behavior of injectiveand at dimension of r-modules under the functors homr(-,-) and -×r-.

We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. This provides some further evidence for the veracity of the Auslander Conjecture. Along the way, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups of π1 to have vanishing...

It will be shown that for any given pure extension A -» B of noetherian k -algebras, with k being a field of characteristic zero, and for any prime ideal b Q A the Zariski-Lipman conjecture for Av is solvable, if £ is a locally factorial domain for which the finite differential module is reflexive. We will also discuss an embedding property with respect to the module of Zariski differentials of...

In this paper, we study the group Pext(C, A) for locally compact abelian (LCA) groups A and C. Sufficient conditions are established for Pext(C, A) to coincide with the first Ulm subgroup of Ext(C, A). Some structural information on pure injectives in the category of LCA groups is obtained. Letting C denote the class of LCA groups which can be written as the topological direct sum of a compactl...