Let $k$ be a field, $A$ a finite dimensional $k$-Algebra, $\text{mod}\,A$ the category of finite dimensional left $A$-modules and $\text{rad}\_A$ the collection of radical morphisms in $\text{mod}\,A$. For a natural number $n$ we denote by $\text{rad}\_A^n$ the collection of $n$ compositions of morphisms in $\text{rad}\_A$ and set $\text{rad}\_A^{\omega} := \bigcap\_{n\in \mathbb{N}} \text{rad}\_A^n$. It is well-known that if $A$ is of infinite representation type, then $\text{rad}\_A^{\omega}\neq 0$. However, in every example it seems to be that we always have $(\text{rad}\_A^{\omega})^2 \neq 0$ aswell.


**Question:** Is it always true that if $A$ is of infinite representation type, then $(\text{rad}\_A^{\omega})^2 \neq 0$?



For a finite dimensional $k$-Algebra $A$ does infinite representation type imply $(\text{rad}_A^{\omega})^2 \neq 0$?
This is proven in the paper (as the title suggests):


*Coelho, Flávio U.; Marcos, Eduardo N.; Merklen, Héctor A.; Skowroński, Andrzej*, [**Module categories with infinite radical square zero are of finite type**](https://doi.org/10.1080/00927879408825084), Commun. Algebra 22, No. 11, 4511-4517 (1994). [ZBL0812.16019](https://zbmath.org/?q=an:0812.16019).


