Finitely generated modules over a PIR

There is the known result that, for two finitely generated modules (f.g.) $M,N$ over a principal ideal domain $R, \ N \subseteq M \implies \mu(N) \leq \mu(M) \ (1)$, where $\mu$ denotes the minimal cardinality of a generating set of a module. It is also easy to show that $(1)$ is not true if R possesses non-principal ideals. However, I was wondering, is $(1)$ true when $R$ is just a principal ideal ring (PIR) and not a domain, think e.g. $\mathbb{Z}$ modulo some composite n? Also, if it is false, is there another property that makes a non-domain PIR satisfy $(1)$?

EDIT: SOLVED, PLEASE IGNORE