@MojtabaJazaeri Take a look at the paper doi.org/10.1006/jabr.1994.1266 the title of the paper is ``On Hadamard Groups"

Thanks! Actually I undertand from your example that I need a notion of ``irreducibilty" on a zero divisor something like the following: the element should not be written as the product of two group algebra element with support size less than the element and maybe this property that (one of them is a zero divisor). I will pose my question for the first unsettled case of existence of zero divisors with respect to the support size.

@SteffenKionke The condition ``residually finite" on a finitely generated group of finite exponent garantees that the group is finite.

@TarasBanakh The following two comments may be relevant: (1) If $G$ is an infinite non-abelian group then $G$ contains infinite subsets $X$ and $Y$ such that $XY\cap YX=\varnothing$. (2) If $a$ is a non-central element of an infinite group $G$, then there exists an infinite subset $X$ of $G$ such that $aX\cap Xa=\varnothing$.

@YCor Could you please give me a reference(or a proof) for your claim: $ZG$ has no zero divisor then $CG$ has no zero divisor?