In this note we show that the main results of the paper [PR] can be obtained as consequences of more general results concerning categories whose morphisms can be uniquely presented as compositions of morphisms of their two subcategories with the same objects. First we will prove these general results and then we will apply it to the case of finite noncommutative sets.

We prove that deformations of tame Krull-Schmidt bimodule problems with trivial differential are again tame. Here we understand deformations via the structure constants of the projective realizations which may be considered as elements of a suitable variety. We also present some applications to the representation theory of vector space categories which are a special case of the above bimodule problems.

Comparing the bounded derived categories of an algebra and of the endomorphism algebra of a given support $\tau$-tilting module, we find a relation between the derived dimensions of an algebra and of the endomorphism algebra of a given $\tau$-tilting module.

We complete the derived equivalence classification of all weakly symmetric algebras of domestic type over an algebraically closed field, by solving the problem of distinguishing standard and nonstandard algebras up to stable equivalence, and hence derived equivalence. As a consequence, a complete stable equivalence classification of weakly symmetric algebras of domestic type is obtained.

We construct derived equivalences between generalized matrix algebras. We record several corollaries. In particular, we show that the $n$-replicated algebras of two derived equivalent, finite-dimensional algebras are also derived equivalent.

In this paper, we introduce related comparability for exchange ideals. Let $I$ be an exchange ideal of a ring $R$. If $I$ satisfies related comparability, then for any regular matrix $A\in M_n\left(I\right)$, there exist left invertible $U_1,U_2\in M_n\left(R\right)$ and right invertible $V_1,V_2\in M_n\left(R\right)$ such that $U_1{V}_{1}A{U}_2{V}_2=diag(e_1,\cdots ,e_n)$ for idempotents $e_1,\cdots ,e_n\in I$.

Let $R⋉M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$ such that $M_R$, ${}_{R}M$, ${(R,0)}_{R⋉M}$ and ${}_{R⋉M}(R,0)$ have finite flat dimensions. We prove that $(X,\alpha )$ is a Ding projective left $R⋉M$-module if and only if the sequence $M{\otimes }_{R}M{\otimes }_{R}X\stackrel{M\otimes \alpha }{⟶}M{\otimes }_{R}X\stackrel{\alpha }{\to }X$ is exact and $\mathrm{coker}\left(\alpha \right)$ is a Ding projective left $R$-module. Analogously, we explicitly describe Ding injective $R⋉M$-modules. As applications, we characterize Ding projective and Ding injective modules over Morita context rings with zero bimodule homomorphisms.

We show here that a directing component of the Auslander-Reiten quiver of a quasitilted algebra is either postprojective or preinjective or a connecting component.

We study the second Hochschild cohomology group of the preprojective algebra of type D₄ over an algebraically closed field K of characteristic 2. We also calculate the second Hochschild cohomology group of a non-standard algebra which arises as a socle deformation of this preprojective algebra and so show that the two algebras are not derived equivalent. This answers a question raised by Holm and Skowroński.