TITLE: Homomorphism between modules and Jacobson radical
QUESTION [2 upvotes]: Let $R$ be a (not necessarily commutative) ring. Let $M$ be a (left) $R$-module. If $N$ is a submodule of $M$, then prove that $$ (N+ \operatorname{rad}(M)) /N \subset \operatorname{rad}(M/N).$$
Here, $\operatorname{rad}(M)$ denotes the Jacobson radical of $M$, i.e. the intersection of all maximal submodules of $M$.

I tried to prove using the fact that:
If $f: A \to B$  is a homomorphism between modules then $f(\operatorname{rad}A) \subset \operatorname{rad} B$ but I didn't succed to construct such a homorphism.

REPLY [1 votes]: Give a maximal submodule $I \subset B$, then $f^{-1}(I)$ is a maximal submodule of $A$, thus $radA \subset f^{-1}(radB)$. So $f(radA) \subset fof^{-1}(radB) \subset radB$.
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