Example of a non noetherian module $M$ s.t. $M/IM$ is noetherian

Question

1. What is an example of a non noetherian module $M$ s.t. $M/IM$ is noetherian?

2. What is an example of a non artinian module $M$ s.t. $M/IM$ is artinian?

What I was thinking that $k[x_1,...,x_n,...]$ is not noetherian but $k[x_1,...,x_n]$ is noetherian and the later is contained in the previous one, so I am done. What about the artinian case?

Answer 1

Take $R$ to be your favorite non-noetherian (hence non-artinian) commutative ring, $M=R$ and $I$ a maximal ideal of $R$.