Let $R$ be a commutative unitary ring and $I$, $J$, $L$ be ideals of $R$ with $L$ proper, $L \subseteq I$ and $L \subseteq J$.
A homework question asks to prove that if $R$ is noetherian then $I$ is a $J$-primary ideal of $R$ (i.e., $I$ is primary and $\sqrt{I} = J$) if and only if $I/L$ is a $J/L$-primary ideal of $R/L$.
I do not understand why assume that $R$ is noetherian. I proved that for each $x \in R$, we have $x + L \in I/L$ if and only if $x \in I$ (also, $x + L \in J/L$ if and only if $x \in J$). Hence, the claim follows from a straightforward application of the definitions of primary ideal and radical of an ideal.
My reasoning is correct? If not, how the hypothesis "$R$ is noetherian" should be used?
Thank you in advance for any suggestion.