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

We prove that the study of the category C-Comod of left comodules over a K-coalgebra C reduces to the study of K-linear representations of a quiver with relations if K is an algebraically closed field, and to the study of K-linear representations of a K-species with relations if K is a perfect field. Given a field K and a quiver Q = (Q₀,Q₁), we show that any subcoalgebra C of the path K-coalgebra K◻Q containing $K^{◻}Q₀\oplus K^{◻}Q₁$ is the path coalgebra $K^{◻}(Q,)$ of a profinite bound quiver (Q,), and the category C-Comod...

Let $𝒢$ be an abstract class (closed under isomorpic copies) of left $R$-modules. In the first part of the paper some sufficient conditions under which $𝒢$ is a precover class are given. The next section studies the $𝒢$-precovers which are $𝒢$-covers. In the final part the results obtained are applied to the hereditary torsion theories on the category on left $R$-modules. Especially, several sufficient conditions for the existence of $\sigma$-torsionfree and $\sigma$-torsionfree $\sigma$-injective covers are presented.

Recently, Rim and Teply , using the notion of $\tau$-exact modules, found a necessary condition for the existence of $\tau$-torsionfree covers with respect to a given hereditary torsion theory $\tau$ for the category $R$-mod of all unitary left $R$-modules over an associative ring $R$ with identity. Some relations between $\tau$-torsionfree and $\tau$-exact covers have been investigated in . The purpose of this note is to show that if $\sigma =({𝒯}_{\sigma },{ℱ}_{\sigma })$ is Goldie’s torsion theory and ${ℱ}_{\sigma }$ is a precover class, then ${ℱ}_{\tau }$ is a precover class whenever...