Reading answers for one of my homework, I found a statement:
Let $R$ be a finite commutative ring, $a \in R$, and $F\colon R \to R$ is given by $F(b) = ab$. If a is not a unit, then $F$ is not surjective. So as $R$ is finite, $F$ is not injective.
How does the implication work here? I understand the definitions of injection and surjection but I find it difficult to apply them here. Why does the finiteness of a ring imply its injection? Hope some experts can help me.
If $\;f\;$ is surjective, then there is $\;b\in R\;$ s.t. $\;ab=1\;$ (assuming $\;R\;$ is unitary) , and thus $\;a\;$ is a unit...contradiction.
And now just observe that a function from any finite set fo itself is injective iff it is surjective.