# Algebra with all modules non-rigid 2

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?

Two other questions:

1. Is it local when it is selfinjective?

2. Is it selfinjective when it is local?

• Same counterexample, just because for the path algebra of $A_2$ there are no non-projective non-injective modules? – Julian Kuelshammer Jul 24 '17 at 8:45
• Ok I added that at least one such module exists. So no hereditary algebras should work. – Mare Jul 24 '17 at 8:49
• Is the title grammatically correct? – Qfwfq Jul 24 '17 at 9:27

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 $\text{Ext}^1_A(S_1,S_1)\neq0$.