Question

(a) If is a unit in ${\mathrm{\mathbb{Z}}}_n$. prove that $a$is not a zero divisor.

(b) If $a$is a zero divisor in ${\mathrm{\mathbb{Z}}}_n$, prove that $a$is not a unit.

Short answer

(a) It is proved that if $a$is a unit in ${\mathrm{\mathbb{Z}}}_n$, then $a$is not a zero divisor.

(b) It is proved that if $a$is a zero divisor, then $a$is not a unit in ${\mathrm{\mathbb{Z}}}_n$.

Step-by-step solution

Part (a)

Assume that $a$is a unit in ${\mathrm{\mathbb{Z}}}_n$, then there will be some $b$so that $ab=1$. Now, if $a$is also a zero divisor, then there exists some $0\ne c\in {\mathrm{\mathbb{Z}}}_n$, such that $ca=0$.

Thus,

$\begin{array}{rcl}c·1& =& c\left(ab\right)\\ & =& \left(ca\right)b\\ & =& 0\end{array}$

But is $c$non-zero, which is a contradiction. Hence, there exists some $c$and $a$are not a zero divisor.

Part (b)

To prove this part, take the contrapositive of part (a). According to the definition of contrapositive, $p\to q$gives role="math" localid="1646390142415" $~q\to ~p$.

According to part (a), if is a unit in ${\mathrm{\mathbb{Z}}}_n$, then is not a zero divisor.

If is a zero divisor, then $a$is not a unit in ${\mathrm{\mathbb{Z}}}_n$, by contrapositive.

Hence proved.