Let $G$ be a finite group and $\mathbb{C}G$ its group ring. Left multiplication by $\alpha\in \mathbb{C}G$ is a linear map $\alpha:\mathbb{C}G \to \mathbb{C}G$, and so $\alpha$ has a left determinant $\det(\alpha)$. If $\alpha\beta = 0$ for some non-zero $\beta$ (i.e. $\alpha$ is a left zero-divisor in $\mathbb{C}G$) then $\det(\alpha)=0$. The converse is also true.

For $G$ abelian $\det(\alpha)$ is zero just in case $\alpha$ lies on a coordinate hyperplane associated with the character idempotents. (This is an easy consequence of this PlanetMath page.)

What is known about the locus $\det(\alpha)= 0$ in $\mathbb{C}G$ when $G$ is finite but not abelian? Are there any similar statements?