Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ an ideal of $R$, $M$ finitely generated $R$--module, and $X$ arbitrary $R$--module. In this paper, for non-negative integers $s, t$ $R$--module $N$, we study the membership $\operatorname{Ext}_{R}^{s}(N, \operatorname{H}^{t}_{\mathfrak{a}}(M, X))$ in Serre subcategories category $R$--modules present some upper bound...

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 ...

Recently, Takahashi established a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies the spherical approximation theorem due to Auslander and Bridger and the Cohen–Macaulay approximation theorem due to Auslander and Buchweitz. In this paper we generalize these results to much more general case over non-commutative rings. As an application, w...

For a unitary right module M , there are two known partitions of simple modules in the category σ[M ]: the first one divides them into M -injective modules and M -small modules, while the second one divides them into M -projective modules and M -singular modules. We study inclusions between the first two and the last two classes of simple modules in terms of some associated radicals and proper ...

We show that every injective homological ring epimorphism f : R → S where SR has flat dimension at most one gives rise to a 1cotilting R-module and we give sufficient conditions under which the converse holds true. Specializing to the case of a valuation domain R, we illustrate a bijective correspondence between equivalence classes of injective homological ring epimorphisms originating in R and...

Let Λ be a preprojective algebra of Dynkin type, and let G be the corresponding semisim-ple simply connected complex algebraic group. We study rigid modules in subcategories Sub Q for Q an injective Λ-module, and we introduce a mutation operation between complete rigid modules in Sub Q. This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to G.

Let Λ be a preprojective algebra of Dynkin type, and let G be the corresponding complex semisimple simply connected algebraic group. We study rigid modules in subcategories Sub Q for Q an injective Λ-module, and we introduce a mutation operation between complete rigid modules in Sub Q. This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to G.

We use Quillen model structures to show a systematic method lift recollements of hereditary abelian categories their associated homotopy categories. To that end, we the notion adjoint triples and investigate transfers along pairs. Applications include liftings module derived counterpart, provide models for stable Gorenstein projective injective modules n-morphism over Iwanaga-Gorenstein rings.

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"...

Motivated by the relation between Schur algebra and group of a symmetric group, along with other similar examples in algebraic Lie theory, Min Fang Steffen Koenig [15], [16] addressed some behaviour endomorphism generator over algebra, which they called gendo-symmetric algebra. Continuing this line works, we classify article representation-finite algebras that have precisely one isomorphism cla...