Quotient ring of a maximal ideal and a subideal

Question

If $R$ is a ring, let $I\subseteq R$ be a maximal ideal. Suppose $J\subseteq I\subseteq R$ is also an ideal. What is the relation between $R/I$ and $R/J$? Like, is $R/I\subseteq R/J$ or vice versa?

Answer 1

$R/I$ is the set of cosets for $I$, and $R/J$ is the set of cosets for $J$. Neither one is a subset of the other.

If you're having trouble visualizing this, try with $R=\mathbb Z$ and $I=(2)$ and $J=(6)$.

The only easy relationship you have is that $(R/J)/(I/J)\cong R/I$.

Perhaps a more interesting observation is that $I/J$ is a maximal ideal of $R/J$, and this is just a reflection of the fact that maximal ideals of $R/J$ correspond to maximal ideals of $R$ containing $J$.

Answer 2

You can say that $I/J$ is a maximal ideal of $R/J$. and that there is a surjective ring homomorphism $R/J\to R/I$ (with kernel $I/J$).

In general you can neither embed $R/I$ into $R/J$ nor $R/J$ into $R/I$. For instance, if $R=\mathbb{Z}$, $I=2\mathbb{Z}$ and $J=\{0\}$.