Local endomorphism rings and indecomposable modules

Question

Given a finite dimensional algebra $A$. For which such algebras is it true that a (not necessarily finitely generated) module $M$ is indecomposable iff its endomorphism ring is local? This is for example true for representation-finite algebras. Are there other examples?

Answer 1

Probably there are no other examples.

The paper

Brenner, Sheila; Ringel, Claus Michael, Pathological modules over tame rings, J. Lond. Math. Soc., II. Ser. 14, 207-215 (1976). ZBL0356.16010.

doesn't explicitly discuss indecomposable modules with non-local endomorphism ring, but does discuss pathological phenomena such as having indecomposable modules $M$, $N_1$ and $N_2$ with $M\oplus N_1\cong M\oplus N_2$, but $N_1\not\cong N_2$, which couldn't happen if $M$, $N_1$ and $N_2$ all had local endomorphism rings.

They remark that such pathologies were known to occur for $k\langle x,y\rangle$-modules and prove that they do for $k[t]$-modules (which also proves it for $k\langle x,y\rangle$-modules, of course).

Therefore, if $A$ is a finite dimensional algebra with a representation embedding of $\text{Mod-}k\langle x,y\rangle$ or $\text{Mod-}k[t]$ into $\text{Mod-}A$ (i.e., an exact functor preserving non-isomorphism and indecomposability), then $A$ must have indecomposable modules with non-local endomorphism rings.

For any wild algebra $A$ there is a representation embedding of $\text{Mod-}k\langle x,y\rangle$ into $\text{Mod-}A$.

In the same paper they also remark that for every known tame algebra $A$ there is a representation embedding of $\text{Mod-}k[t]$ into $\text{Mod-}A$. But that was 1976, and maybe more is known now.