Let $R$ be a (commutative, unital) ring. Define the norm of a nonzero ideal $I$ of $R$ to be $$N(I) = \# R/I$$ which we take to be $\infty$ if the quotient ring is infinite. Also define $N(0) = 0$.
I want to prove $$N(IJ) \ge N(I)N(J)$$ for all ideals $I, J$ of $R$.
Attempt
Given $I, J$, then $IJ \subseteq I$ and $IJ \subseteq J$, so $IJ \subseteq I \cap J$. Therefore the surjective map $r \mapsto r + I \cap J$ descends to the quotient $R/IJ$, and we have a surjection $$R/IJ \twoheadrightarrow R/(I \cap J).$$ The latter ring is isomorphic to $R/I \times R/J$ because $I \cap J$ is the kernel of $r \mapsto (r + I, r + J)$. Comparing cardinalities, it follows that $N(IJ) \ge N(I) N(J)$.
Question
I have doubts because this paper by S. Marseglia https://arxiv.org/abs/1810.02238 studies super-multiplicativity of $N$ as a non-trivial property that only certain rings possess. What's going on? Is there an error in my proof?