# All possible zero divisors of the ring $\Bbb Z_n \oplus \Bbb Z_m$.

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.

• In general, the zero divisors $(\alpha, \beta) \in \mathbb{Z}_{n} \oplus \mathbb{Z}_{m}$ can be described as elements where at least one of $\alpha, \beta$ is a zero divisor (or indeed simply $0$) and $\alpha$ and $\beta$ are not both zero. Feb 12, 2015 at 2:27
• @AlexWertheim Maybe you find this helpful: en.wikipedia.org/wiki/Zero_divisor#Zero_as_a_zero_divisor Feb 12, 2015 at 7:59
• @user26857: forgive me, I am accustomed to the convention where $0$ is excluded as a zero divisor. Thanks for sharing. Feb 12, 2015 at 15:09

Write $(x,y)(z,w) = 0$, so $xz = 0$ mod $n$ and $yw = 0$ mod $m$. Now solve for $x,y,z,w$.