There is a proposition on page 390 of Hungerford's ``algebra'' (1974) as follows:
Proposition 4.6. Let $J$ be an ideal in a commutative ring $R$ with identity. Then $J$ is contained in every maximal ideal of $R$ if and only if for every $R$-module $A$ satisfying the ascending chain condition on submodules, $\cap_{n = 1}^{\infty} J^{n}A = 0$.
To prove the ($\Leftarrow$) part, the author picks any maximal ideal $M$ of $R$ and uses the fact that $A = R / M$ is a nonzero $R$-module that has no proper submodules. Then he goes on to argue that since $JA$ is a submodule of $A$, either $JA = A$ or $JA = 0$ and then makes a reasoning that $JA = A$ results in a contradiction and hence $JA = 0 = J(R/M)$ which, in turn, implies that $J \subset JR \subset M$ and so completes the proof.
At this point, I can't seem to understand the inclusion $J \subset JR$.
Any hints?
This is just a tautology : $J$ is an ideal of $R$ so $J = JR$.
(Precisely, $JR\subset J$ because $J$ is an ideal, and $J\subset JR$ because if $j\in J$ then $j = j\cdot 1\in JR$.)