Prove the Zero divisor of Z[2]

Question

$K$ is a field and $A=K[\mathbb Z/2\mathbb Z\times\mathbb Z/2\mathbb Z]$ is a group algebra from $\mathbb Z/2\mathbb Z\times\mathbb Z/2\mathbb Z$ to $K$. How can I prove that there is a zero divisor in $A$? Shall I find the basis first?

Answer 1

Take $g \in \mathbb Z/2\mathbb Z \times \mathbb Z/2\mathbb Z$ to be $g=(1,0)$. Then, $g^2=1$, so in the group algebra $g^2-1=0$, which is $(g+1)(g-1)=0$. Thus, $g+1$ and $g-1$ are zero divisors.