Given the ring $ \mathbb{Z}/n\mathbb{Z} $ is always true that $ \mathbb{Z}/n\mathbb{Z}=[\text{zero divisors}]\cup[\text{units}] $
How can evaluate the zero divisors and units ?
I believe that $ a x=0 \pmod n $ for zero divisor
and $ ax=1 \pmod n $ for units
I know how to solve a congruence but what is $a$ ?? thanks.
What are the generators of the group $ \mathbb{Z}/n\mathbb{Z} $ under the addition '+' and product '$\times$'?
Best Answer
Hint $ $ If $\rm\,a\in\rm\color{#c00}{finite}$ ring $\rm\,R\,$ then $\,\rm x\to ax\,$ is onto $\rm\color{#c00}{iff}\ 1$-$1,\,$ so $\rm\,a\,$ is a unit iff $\rm\,a\,$ is not a zero-divisor.