Given a finite dimensional connected algebra $A$ with $Ext^{1}(M,M) \neq 0$ for any non-projective and non-injective indecomposable module $M$, with the condition that at least one such module exists.


1. Is $A$ selfinjective?
2. Is $A$ local?


(answer is no,see the answer by Jeremy Rickard)


Two other questions:


3. Is it local when it is selfinjective?
4. Is it selfinjective when it is local?



Algebra with all modules non-rigid 2
Let $A=kQ/\text{rad}(kQ)^2$, where $Q$ has two vertices, a loop at vertex $1$ and an arrow from vertex $1$ to vertex $2$. I think it has five indecomposable modules, all of which are either projective or injective apart from the simple $S\_1$ at vertex $1$, and $\text{Ext}^1\_A(S\_1,S\_1)\neq0$.


