Show that (ZZ_6 +_6, xx_6) is a commutative ring. Is (ZZ_6 +_6, xx_6) a field?

Since addition modulo n and multiplication modulo n of integers are commutative . So, the ring ${\displaystyle ({\mathbb{Z}}_6{+}_6,{\times }_6)}$ is a commutative ring.
Suppose ${\displaystyle a,b\in {\mathbb{Z}}_6}$
Then, $ab=ba=m$ , where m is the reminder obtained when ab, ba is divided by 6 and ${\displaystyle 0\le m\le 5}$
ab ${\displaystyle \equiv }$ $m\mathrm{mod}6,ba$ ${\displaystyle \equiv }$ $m\mathrm{mod}6$
Also, $a+b=b+a=n$ , where n is the reminder obtained when $a+b,b+a$ is divided by 6 and ${\displaystyle 0\le n\le 5}$
$a+b$ ${\displaystyle \equiv }$ $n\mathrm{mod}6,b+a$ ${\displaystyle \equiv }$ $n\mathrm{mod}6$
Hence, ${\displaystyle {\mathbb{Z}}_6}$ is a commutative ring
A ring is a field if its every element has an inverse.
As ${\displaystyle 2,3\in {\mathbb{Z}}_6}$
but $2\cdot 3=6\mathrm{mod}6=0$
So, 2 and 3 are zero divisors and do not have inverse .
So, ${\displaystyle ({\mathbb{Z}}_6{+}_6,{\times }_6)}$ is not a field.