What can be the all possible zero divisors of the ring $\Bbb Z_n \oplus \Bbb Z_m$, where $n$ and $m$ belongs to $\Bbb N$??
One can easily verify that $(\bar a,\bar0)$ and $(\bar 0 , \bar b)$ are zero divisors!! Moreover when $gcd(n,m) \neq 1$ there are also other types of zero divisors than of the form $(\bar a,\bar0)$. For example, when we consider $\Bbb Z_4 \oplus \Bbb Z_6$ then we see that $(\bar 2, \bar3)$ and $(\bar 2, \bar 2)$ are zero divisors.