Section 2.3: The Arithmetic of $latex\mathbb{Z}_p$ (p Prime) and $latex\mathbb{Z}

In my analysis of this section I noted that Hungerford often switches between the conventional modular arithmetic notation (with the brackets []) and then uses the number by itself. While both are completely valid certain uses are better than others. That is the notation $[a]$ is much better than a if there are integers in being used at the same time. That is if $b\in\mathbb{Z}$ and $a\in\mathbb{Z}_n$ then $ab$ and $b\cdot [a]$ mean the same thing but the second notation indicates that b is not a congruence class.

Now to continue with Section 2.3:

Question:

Does this work for $\mathbb{Z}_13$ ? $[a]\odot [b]=[ab]=[0]$ means $13\mid ab$ implies $13\mid a$ or $13\mid b$ .

We’ll later cover this but note that if we try to solve $ax=1$ in $\mathbb{Z}$ then the only solutions for a are $a=\pm 1$ . But the same equation in $\mathbb{Z}_5$ has multiple solutions: $[2]\odot [3]=[1]$ and $[4]\odot [4]=[1]$ .

Theorem 2.8:

For a positive integer p, the following are equivalent:

p is prime

For non zero $a\in\mathbb{Z}_p$ , the equation $ax=1$ has a solution

Whenever $bc=0$ in $latex \mathbb{Z}_p$ then $$b=0$$ or $c=0$