What is the difference between submodules of $A/\mathfrak a$ as $A$-module or as $A/\mathfrak a$-module?

Question

If $\mathfrak a$ is an ideal of unital commutative ring $A$, then we can consider $A/\mathfrak a$ as $A$ module or as $A/\mathfrak a$ module.
If $A=\mathbb Z$ there is no structural difference between submodules of them, but is it true in general? For example submodules in both cases are in the one-to-one correspondence?

Answer 1

An example of a difference: $\mathbb{Z}/3\mathbb{Z}$ is projective (and free) as a $\mathbb{Z}/3\mathbb{Z}$-module but is not projective (nor free) as a $\mathbb{Z}$-module.

However things like simplicity and indecomposability will remain the same, which is easy to prove since the action of $A/\mathfrak a$ is given by the action of $A.$

The only thing you're really changing is which category you are viewing the module in, so it shouldn't be surprising that categorical notions like projectivity might change while the structure won't.

EDIT

I missed the point that you are really concerned with submodules (which is structural and won't change).

The action of $A/\mathfrak{a}$ on itself is given by $(a\mathfrak{a})\cdot(b\mathfrak{a}) = ab\mathfrak{a}.$ The action of $A$ on this module is $a\cdot(b\mathfrak{a}) = ab\mathfrak{a}.$ Both actions are well defined (check).

Then if $M$ is a submodule of $A/\mathfrak{a}$ as an $A$-module then $aM\subset M$ for all $a\in A.$ Thus the $(a\mathfrak{a})M = aM\subset M$ and hence $M$ is a submodule of $A/\mathfrak{a}$ as an $A/\mathfrak{a}$-module. The converse is similar.