Quotient rings (modules) of a bigger ideal is contained in the quotient ring (modules) of a smaller ideal?

Question

I have a question about rings and then about general modules. Suppose $R$ is a ring and $I$, $J$ are proper ideals of $R$ such that $J\subset I$. Then my first question is “is $R/I\subset R/J$ as quotient rings?” What about as submodules? I.e. is $R/I$ a $R$-submodule of $R/J$?

Answer 1

No, $R/I$ cannot be in general identified with a subring of $R/J$.

Consider the case $R=\mathbb{Z}$, $I=2\mathbb{Z}$ and $J=6\mathbb{Z}$. Then $R/I$ and $R/J$ are the rings of residue classes modulo $2$ and $6$ respectively, which have no proper subring.

What's generally true is that $R/I$ is a quotient ring of $R/J$, which is precisely the statement of the third (but the numbering depends on the textbook) homomorphism theorem: in this case $I/J$ is an ideal in $R/J$ and the quotient ring $(R/J)/(I/J)$ is isomorphic to $R/I$.

The same goes for the module case. The example $\mathbb{Z}$ can be misleading, because in the case of a PID, the quotient $R/I$ is isomorphic to a submodule of $R/J$ (use uniqueness of factorization). But in general this does not happen.