How do you calculate this product $\mathbb{Z}_6\times\mathbb{Z}_6$

Question

How do direct product (Cartesian product) of $$\mathbb{Z}_6\times\mathbb{Z}_6$$

I need to know if this product is a direct integral domain (ring integrity).

Answer 1

write $Z_6$ as ${0,1,2,3,4,5}$ then $\mathbb{Z}_6\times\mathbb{Z}_6$ looks like {$(0,0) , (0,1) , (0,2)...(0,5), (1,0), (1,1), ... (1,5), (2,0)...(2,5), (3,0),...(3,5) .... (5,5)$} in total $36$ such tuples.

Answer 2

The best answer that I can give you is this:

http://en.wikipedia.org/wiki/Direct_product_of_groups