# For what modules is the endomorphism ring a division ring?

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 $$\text{End}(M)$$ is a division ring? Notice if $$M$$ is simple, then it happens and if it happens, then $$M$$ but be indecomposable. So this property is something inbetween simple and indecomposable, and it came up in the context of torsion pairs.

• When $A$ is an arbitrary commutative (associative unital) ring, it's equivalent to being simple. Indeed, in this case, every length 1 submodule of $M$ is isomorphic to a quotient of $M$. – YCor Nov 26 '20 at 11:58
• If $A$ is the algebra of upper triangular $2\times 2$ matrices and $M=k^2$, then $M$ has this property (although it's not simple). This is because the centralizer of $A$ in $M_2(k)$ is reduced to scalars, and this is precisely the endomorphism $k$-algebra of the $A$-module $M$. – YCor Nov 26 '20 at 12:01
• Crosspost – rschwieb Dec 1 '20 at 15:36