Let $R$ be a commutative rng, i.e. a commutative ring without an identity element.
Does $R$ still have the Invariant Basis Number (IBN) property?
Recall that a ring is said to have the IBN property if $R^m \cong R^n \Rightarrow m=n$.
All commutative rings have the IBN property, but the standard proof I know makes an essential use of the existence of a maximal ideal by Zorn's Lemma and passing to the residue field, which depends on the presence of a unit.