Let $\mathbb{K}$ be a field and consider the ring of dual numbers $R=\mathbb{K}[x]/<x^2>$. I have to prove that any projective $R$-module $P$ is injective.
My idea is to use the Baer’s criterion.
The ideal of $R$ are: $0$ , $<\bar{x}>$, $R$. So working with $<\bar{x}>$ is enough.
Let $\phi: <\bar{x}> \to P$ a morphism of $R$-module. I have to extend it to a morphism from $R$ to $M$, the problem is where I have to send the unit of $R$. I have no idea how to use the hypothesis.