6
$\begingroup$

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?

  • 0
    Do you have a particular $\phi$ in mind? If so, which one?2011-04-02
  • 0
    No, I don't. I'm just trying to find the complete solution for the first case.2011-04-03

2 Answers 2