Let $\mathbb{Z}$ be the integers.
The set $M$ of all $\mathbb{Z}$-module homomorphisms from a ring $R$ with unity to a divisible abelian group $A$ is known to be an injective left $R$-module.
I am wondering if the word "divisible" for $A$ dropped, is $M$ still injective? Is there any concrete example or source for me to read to find the answer? Thanks in advance.
No chance, take $\mathbb{Z}$-modules and consider $\mathbb{Z}/n\mathbb{Z}$ then your $M$ would be $\mathbb{Z}/n\mathbb{Z}$, which is faaaaar from being injective for $n\neq 0,1$.