$\varphi : R\rightarrow \mathbb{Z}/p\mathbb{Z},x\mapsto x+p\mathbb{Z}$ is well-defined and surjective.

Question

Given the ring $$R=\{x\in\mathbb{Q}\mid x=\frac{a}{b}, a.b\in\mathbb{Z}, p\nmid b\}$$ for a prime number $p$.

Now I have to show that $$\varphi : R\rightarrow \mathbb{Z}/p\mathbb{Z},x\mapsto x+p\mathbb{Z}$$

is well-defined and surjective.

Until now I just thought that $\varphi$ could be surjective because $x=\frac{a}{b}$ and the map $a\mapsto a+p\mathbb{Z}$ is surjective. But from here on I don't really know where to go next.

Answer 1

The division by $b$ in $R$ corresponds to a multiplication by $b^{-1}$ in $\mathbb Z/p\mathbb Z$; $b^{-1}$ is well-defined since $p$ is prime (thus $\mathbb Z/p\mathbb Z$ is in fact a field) and $p\nmid b$, so $\varphi$ is well-defined.

$\varphi$ is surjective because $\frac01,\frac11,\dots,\frac{p-1}1$ are all in $R$ and naturally map to each distinct element in $\mathbb Z/p\mathbb Z$.