Product of nilpotent ideal and simple module is zero

Question

I am stuck with trying to show that if an ideal $I$ of a ring $R$ is nilpotent and $M$ is a simple $R$-module, then $IM = 0$.

I have attempted showing this by using the fact that the annihilator of a simple module is the primitive ideal, and I'm guessing trying to show that a nilpotent ideal and a primitive ideal are some how related but i think i am missing some crucial information.

I have tried using properties of maximal ideals but to no conclusion, I'm sure I'm just missing an initial step any help on this will be greatly appreciated

thanks in advance

Answer 1

The connection you are looking for is that nilpotent ideals are all contained in the Jacobson radical. This is easy to see since the primitive ideals of a ring are prime, and hence each one has to contain all nilpotent ideals. Thus their intersection (the Jacobson radical) contains all nilpotent ideals.

Since the Jacobson radical annihilates simple $R$ modules, so must each nilpotent ideal.

Answer 2

Let $M$ be a non-zero simple module. As $M$ is simple, $IM = 0$ or $IM = M$. If $IM = M$, then note that $IM = I^{n}M$ $\forall n \geq 1$. But as $I$ is nilpotent, $I^k = 0$ for some $k$. This implies that $M = 0$, a contradiction. Notice that a consequence is that Nilpotent ideals are contained in the Jacobson radical.