I'm aware of the fact that when a commutative ring $R$ is viewed as a module over itself, the submodules of $R$ are just the ideals. We can pick an ideal $I \subset R$ and take the quotient ring $R/I$, which is also an $R$-module. Are the $R$-submodules of $R/I$ again the ideals of $R/I$ as a ring?
Submodules of a Quotient Ring Viewed as a Module
commutative-algebramodulesring-theory
Best Answer
Yes, this is true. For any $R/I$-module $M$, the $R/I$-submodules of $M$ coincide with the $R$-submodules of $M$. This is because an $R/I$-module is the same thing as an $R$-module killed by $I$ (the $R/I$-module structure is uniquely determined by the $R$-module structure).