Equivalent condition to exactness

Question

Let $I$ be an injective $R$-module. Is it true that a short sequence of $R$-modules $$0\rightarrow A'\rightarrow A\rightarrow A''\rightarrow 0$$ is exact if and only if $$ 0\rightarrow \text{Hom}_R(A'',I)\rightarrow\text{Hom}(A,I)\rightarrow\text{Hom}(A',I)\rightarrow 0$$ is exact? I'm interested particularly in the case $R =\mathbb{Z}$ and $I = \mathbb{Q}/\mathbb{Z}$. Obviously the forward direction follows directly from $I$ being injective, I'm curious about the converse.

Answer 1

Injective modules that satisfy this property are called faithfully injective, which is equivalent to being an injective cogenerator in the category of $R$-modules.

Since $\mathbb{Q}/\mathbb{Z}$ is faithfully injective over any ring, it holds over $\mathbb{Z}$.

More generally, one can consider what are called faithfully exact functors, which are functors $T:R\text{-Mod}\to S\text{-Mod}$ such that $T(A)\to T(B)\to T(C)$ is exact if and only if $A\to B\to C$ is exact. You are just considering the case of the functor $\text{Hom}(-,I)$.

The original paper on this subject is by Ishikawa, and it is very readable and comprehensive, including giving a treatment of faithfully injective modules.