Meaning of $Z\oplus Z$

Question

I am a beginner in Ring Theory and just started Integral Domains. In my textbook, the following was stated :

$Z\oplus Z$ is not an integral domain.

I can't understand this. I know $\oplus$ denotes direct sum, but what exactly does direct sum mean? I had thought that $Z\oplus Z$ would mean all points in 2D space with integer coordinates, but in that case I feel $Z\oplus Z$ has

1. commutativity.

2. unity namely $(1,1)$

3. no zero divisor

Hence should be an integral domain.

I am new so please correct me if I am wrong instead of downvoting..

Answer 1

The notation $\Bbb Z\oplus\Bbb Z$ is probably referring to the direct product ring $\Bbb Z\times\Bbb Z$. The wikipedia entry for direct sum advises against using the notation $R\oplus S$ for rings in favor of the more standard $R\times S$.

Whatever notation we choose, we can still show $\Bbb Z\oplus\Bbb Z$ is not an integral domain. Note that $(1,0),(0,1)\in\Bbb Z\oplus\Bbb Z$ but $$ (1,0)\cdot(0,1)=(1\cdot 0, 0\cdot 1)=(0,0) $$ Hence $\Bbb Z\oplus\Bbb Z$ is not an integral domain.