Let $k$ be a field, $A$ a $k$-algebra of finite length and $M$ an $A$-module of finite length. When does it happen, that $\mathrm{End}(M)$ is a division ring? Notice if $M$ is simple, then it happens and if it happens, then $M$ must be indecomposable. So this property is something in between simple and indecomposable, and it came up in the context of torsion pairs.



For what modules is the endomorphism ring a division ring?
Such modules are called bricks for finite dimensional algebras and there are in general very many of them.


Having a division ring as the endomorphism ring is equivalent to the condition that every non-zero endomorphism morphism is invertible.


For hereditary and tilted algebras they are quite interesting, see <https://link.springer.com/article/10.1007/BF03323325> where the name brick might also appeared first.


A classification for given classes of such algebras is an interesting problem, see for example <https://arxiv.org/pdf/1712.08311.pdf> .


Theorem 1.4 in <https://arxiv.org/pdf/1503.00285.pdf> gives a characterisation when an algebra has only finitely many bricks.


