[Math] Find all units and zero divisors

Question

Consider the ring $\mathbb Z_3 \oplus \mathbb Z_6$. Find all units and zero divisors. There are only $4$ units:

$(1,1)(1,1)= (1,1)$

$(1,5)(1,5)=(1,1)$

$(2,1)(2,1)=(1,1)$

$(2,5)(2,5)=(1,1)$.

Answer 1

Hints:

$(0,?)(?,0)=(0,0)$

$2\cdot3=0$ in $\Bbb Z_6$.

To make sure you didn't miss any, count up how many elements you discovered this way and compare it with how big you expect the ring to be.