Why can a field not be used to make a quotient ring?

Question

In socratica's youtube video (which should be required for every abstract algebra class), she is explaining that a motivation for modules as rings allow you to do things that you can't do with fields. If you have a ring with an ideal $I\triangleleft R$ where M is an R module you can create an R/I quotient ring. Why can't you do this with a field?

If $F$ is a field, the only ideals are ${0}$ and $F$. You can do it, but you don’t get anything interesting: just $F$ again, or the trivial module. — Arturo Magidin, Sep 13 '20 at 04:37

score 4 · Accepted Answer · answered Sep 13 '20 at 04:42

If $F$ is a field, its only ideals are $\{0\}$ and $F$. You can do it, but you either get the trivial ring/module or $F$ again. So it is not interesting to do it for fields. It’s not that you can’t do it, but rather that there is no point in doing it.

Why can a field not be used to make a quotient ring?

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