Related papers: Silting Modules over Triangular Matrix Rings

Let $k$ be a field and let $\Lambda$ be a finite dimensional $k$-algebra. We prove that every bounded complex $V^\bullet$ of finitely generated $\Lambda$-modules has a well-defined versal deformation ring $R(\Lambda,V^\bullet)$ which is a…

Let $\Gamma$ be a cancelation monoid with the neutral element $e$. Consider a $\Gamma$-graded ring $A=\oplus_{\gamma\in\Gamma}A_{\gamma}$, which is not necessarily commutative. It is proved that $A_e$, the degree-$e$ part of $A$, is a local…

For four wide classes of topological rings $\mathfrak R$, we show that all flat left $\mathfrak R$-contramodules have projective covers if and only if all flat left $\mathfrak R$-contramodules are projective if and only if all left…

We establish a foundational homotopical framework for ternary $\Gamma$-modules by establishing that $\mathcal{T}\text{-Mod}$ is a Barr-exact, monoidal closed category. We resolve the long-standing "additivity obstruction" in non-binary…

We introduce a general version of singular compactness theorem which makes it possible to show that being a $\Sigma$-cotorsion module is a property of the complete theory of the module. As an application of the powerful tools developed…

Let $R$ denote a Noetherian ring and an ideal $J \subset R$ with $U = \operatorname{Spec R} \setminus V(J)$. For an $R$-module $M$ there is an isomorphism $\Gamma(U, \tilde{M}) \cong \varinjlim \operatorname{Hom}_R(J^n,M)$ known as…

This paper develops the homological backbone of the theory of non-commutative $n$-ary $\Gamma$-semirings. Starting from an $n$-ary $\Gamma$-semiring $(T,+,\tilde{\mu})$ and its $\Gamma$-ideals, we work in the slot-sensitive categories of…

Let $\Lambda$ be a left and right noetherian ring and $\mod \Lambda$ the category of finitely generated left $\Lambda$-modules. In this paper we show the following results: (1) For a positive integer $k$, the condition that the subcategory…

We study maximal almost rigid modules over a gentle algebra $A$. We prove that the number of indecomposable direct summands of every maximal almost rigid $A$-module is equal to the sum of the number of vertices and the number of arrows of…

Given a topological ring $R$, we study semitopological $R$-modules, construct their completions, Bohr and borno modifications. For every topological space $X$, we construct the free (semi)topological $R$-module over $X$ and prove that for a…

Let $\Sigma_r$ be the symmetric group acting on $r$ letters, $K$ be a field of characteristic 2 and $\lambda$ and $\mu$ be partitions of $r$ in at most two parts. Denote the permutation module corresponding to the Young subgroup…

For any ring $R$, we investigate balanced pairs of classes of modules and their relations to cotorsion triples. We characterize the case when a balanced pair generates a tilting cotorsion pair, and dually, when it cogenerates a cotilting…

In this article, the projectivity of finitely generated flat modules of a commutative ring are studied from a topological point of view. Then various interesting results are obtained. For instance, it is shown that if a ring has either a…

Suppose that $G$ is a finite group and $k$ is a field of characteristic $p>0$. We consider the complete cohomology ring $\mathcal{E}_M^* = \sum_{n \in \mathbb{Z}} \widehat{Ext}^n_{kG}(M,M)$. We show that the ring has two distinguished…

In this paper we study the lower triangular matrix $\mathbb{K}$-algebra $\Lambda:=\left[\begin{smallmatrix} T & 0 \\ M & U \end{smallmatrix}\right],$ where $U$ and $T$ are basic $\mathbb{K}$-algebras with enough idempotents and $M$ is an…

Given an associative, not necessarily commutative, ring R with identity, a formal matrix calculus is introduced and developed for pairs of matrices over R. This calculus subsumes the theory of homogeneous systems of linear equations with…

Let $\Lambda$ and $\Gamma$ be left and right noetherian rings and $_{\Lambda}U$ a Wakamatsu tilting module with $\Gamma ={\rm End}(_{\Lambda}T)$. We introduce a new definition of $U$-dominant dimensions and show that the $U$-dominant…

Let $Q$ be a Noetherian ring with finite Krull dimension and let $\mathbf{f}= f_1,... f_c$ be a regular sequence in $Q$. Set $A = Q/(\mathbf{f})$. Let $I$ be an ideal in $A$, and let $M$ be a finitely generated $A$-module with $\projdim_Q…

In this paper we prove that if R is a left Noetherian and left regular ring such that all finitely generated projective left R-modules are stably free, then the same is true for the completion R[[x;\sigma,\delta]] of any Ore extension…

We call a right module $M$ (strongly) virtually regular if every (finitely generated) cyclic submodule is isomorphic to a direct summand. $M$ is said to be completely virtually regular if every submodule is virtually regular. In this paper,…