Question

If $a$is a unit and $b$is a zero divisor in${\mathrm{\mathbb{Z}}}_n$ , show that is a zero divisor.

Short answer

It is proved that $ab$is also a zero divisor.

Step-by-step solution

Use the given part

It is given that $a$be a unit and $b$a zero divisor in ${\mathrm{\mathbb{Z}}}_n$, then for some $0\ne c\in {\mathrm{\mathbb{Z}}}_n$

Proof

As $b$are zero divisor in${\mathrm{\mathbb{Z}}}_n$ , then for some $0\ne c\in {\mathrm{\mathbb{Z}}}_n$,$bc=0$ , then we have the following:

$$\begin{gathered} a\left(bc\right)=0 \\ \left(ab\right)c=0 \end{gathered}$$

It is given that $a$and role="math" localid="1646384116301" $b\ne 0$, then $ab\ne 0$.

Hence, $ab$is also a zero divisor.