The following question has been lingering in my mind for months.
Let $R$ be a non-zero commutative ring with $1$. Consider $\phi : R^n \rightarrow R^m$,
1) as an injective $R$-module homomorphism.
2) as an injective ring homomorphism. (by definition $\phi(1)=1$.)
In which of the above cases, we can deduce that $n \leq m$? and why?