$J/I\trianglelefteq R/J$ Ideal

Question

Let $R$ be a ring and $I,J\trianglelefteq R$ and $I\subset J$. We define $f: R/I \rightarrow R/J, x\mapsto [x]$ Show that:

$J/I\trianglelefteq R/I$ where $J/I = \{[x]\in R/I: x\in J\}$

So I'm supposed to show that this is an ideal. That 0 is contained in it is clear to me, but the fact that we are in a quotient group makes it hard for me to see how to prove the other two criteria for $J/I$ being an ideal.

Answer 1

Let $\pi:R\to R/I$ denote the canonical projection modulo $I$. Then $\pi$ is an $R$-module homomorphism. Moreover, $J/I$ is the image of $J$ through $\pi$ hence $J/I$ is an $R$-submodule of $R/I$. Since $I\subseteq\operatorname{Ann}_R(J/I)$, it follows that $J/I$ is, in fact, an $R/I$-submodule of $R/I$, that's an ideal of $R/I$.