Existence of small projective dimensioned modules

Question

Suppose $A$ is a (if necessary unital) associative ring and $I$ is a left ideal in $A$. Let $\operatorname{pd}(M)$ denote the projective dimension of a left $A$-module $M$.

Then do either of the following exist:

1. $M\in \space _A\mathrm{Mod}$ such that $ \infty >\operatorname{pd}(M)\geq \operatorname{pd}(M/IM)$ and $M\neq 0$?
2. $N\in \space _A\mathrm{Mod}$ such that for all $J \lhd A$, the equation $\infty >\operatorname{pd}(N)\ \geq \operatorname{pd}(N/JN)$ holds and $N\neq 0$?

Answer 1

If $A$ is a commutative local artinian ring which is not a field, and $I \neq 0$ is the maximal ideal in $A$, then no such modules exist.

In this case every $A$-module has projective dimension $0$ or $\infty$, see for instance (Bass, Hyman. Finitistic dimension and a homological generalization of semi-primary rings). By a theorem of Kaplansky, any projective module over a local ring is free.

Suppose $M$ is a non-zero $A$-module with $\operatorname{pd} (M)=0$. Since $I$ is nilpotent, we cannot have $IM=M$. Then $M/IM$ is a non-zero module annihilated by $I$, so it cannot be free. Therefore $\operatorname{pd} (M/IM)=\infty$.