Zero divisors in $(\mathbb Z_n,+,*)$

Question

How to understand this :
An element $a$ in $(\mathbb Z_n,,+,*)$ is a zero divisor iff $a$ and $n$ aren't coprime...

---

EDIT: Is it also true that an element a in $(\mathbb Z_n,,+,*)$ is a unit iff $a$ and $n$ are coprimes..

Answer 1

The way to understand it is this: If $a$ and $n$ has no common prime factors, then the only $b$ so that $ab = 0$ is $0$ itself. However, if $a$ and $n$ do have prime factors in common, then there are other $b$ that does the same.

Example: In $\Bbb Z_{12}$, take $a = 5$. Then, since $\gcd(5, 12) = 1$, there is only one $b$ so that $5b \equiv 0$. However, taking $a = 8$, then since $\gcd(8, 12) = 4 \neq 1$, there are other elements (in fact there are $4$, counting $0$) that annihilate $a$. They are $3$, $6$ and $9$. Therefore $8$ is a zero divisor in $\Bbb Z_{12}$.

And yes, in $\Bbb Z_{n}$, every element is either a zero divisor or a unit, i.e. division is possible if both numerator and denominator are coprime to $n$.