Suppose R is an integral domain with |R|=6. By the definition, R contains two element 1 and 0 where 1 is the multiplication identity and 0 is the addition identity and 1 is not equal to 0. R does not have zero factors.
We consider its zero idea Z=(0).
If Z=(0) is a maximal ideal of R, ...
This is a proof regarding integral domain with six elements.