I am asked to provide an example of a ring $(R,+,\cdot)$ and a Abel group $(M,+)$ such that when $\cdot$ is provided for $M$, an $R$-module can not be formed.
A bit strange question considering, that the definition of a module says that the ring must be unit-ring, such that $0_R\neq 1_R$. So if I construct such a ring, an $R$-module can not be formed.
However, perhaps that's not the intention here, in which case, how could I construct an example provided the ring satisfies the conditions?
Let $R=\mathbb Z/2\mathbb Z$ and $M=\mathbb Z$. What are the possible homomorphisms $\mathbb Z/2 \mathbb Z \to \mathrm{End}(\mathbb Z) \cong \mathbb Z$?