Properties of the $R$-module $M$ and $R/I$-module $M$.

Question

Let $R$ be a commutative ring with $1$ and $M$ is an $R$-module.

Let $I$ be an ideal of $R$ contained in $\mathrm{Ann}(M)$.

Consider the $R/I$-module $M$.

I know the fact that the $R/I$-module $M$ is noetherian (artinian) iff the $R$-module $M$ is noetherian (artinian) because of the property that is submodule is still submodule when we change rings.

Here is my question:

Is that true that the $R$-module $M$ has property $P$ iff the $R/I$-module $M$ has property $P$?

From my intuition, I would say that is true but can't find an explicit proof. I looking for the proof or a counterexample.

Thanks in advance.

Answer 1

In general no. For a counterexample, consider an $R$-module $M$, such that $\mathrm{Ann}_R(M)\neq\{0\}$, that is a non-faithful $R$-module. Then, if you take $I=\mathrm{Ann}_R(M)$ you will get a faithful $R/I$-module $M$, i.e., one for which $\mathrm{Ann}_{R/I}(M)=\{0\}$.