Exercise 2.11 Atiyah-Macdonald [duplicate]

This is part of the exercise, I'm stuck with it.

$A$ is a commutative ring with unit.

1) Suppose we have an homomorphism $\phi : A^{m} \to A^{n}$ surjective. Is true that $m \geq n $ ?

2) Suppose we have an homomorphism $\phi : A^{m} \to A^{n}$ injective. Is true that $m \leq n $ ?

Solution 1:

Hints: The answer to both is yes. For (1) tensor by the residue field of a maximal ideal.

For (2) assume, by way of contradiction, that $m > n$. Consider $A^n \subseteq A^m$ and consider $\phi\colon A^m \to A^m$ as a map into $A^m$. Let $\pi\colon A^m \to A^m$ be a projection map such that $\pi\phi = 0$. Now use Proposition 2.4 on $\phi$ so show it satisfies some relation. Then use $\pi$ and the injectivity of $\phi$ to reduce that relation to $\phi = 0$.