Let G be an abelian group with torsion and C[G] be the group ring over complex numbers C. Is there a clear description or classification of zero divisors of C[G]?
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$\begingroup$ Have you tried writing $\mathbb{C}[G]$ as $\mathbb{C}[x_{1},\dots,x_{r+s}]/\langle x_{1}^{a_{1}}-1,\dots,x_{r}^{a_{r}}-1\rangle$ when $G$ is finitely generated? $\endgroup$– Joseph Van NameCommented Oct 16, 2021 at 14:17
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$\begingroup$ And the ideal $\langle x_{1}^{a_{1}}-1,\dots,x_{r}^{a_{r}}-1\rangle$ in $\mathbb{C}[x_{1},\dots,x_{r+s}]$ is precisely the set of all polynomials $p$ such that $p(u_{1},\dots,u_{r},x_{r+1},\dots,x_{r+s})=0$ whenever $u_{1}^{a_{1}}=\dots=u_{r}^{a_{r}}=1$. $\endgroup$– Joseph Van NameCommented Oct 16, 2021 at 15:45
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$\begingroup$ @JosephVanName, what if the group is not finitely generated? $\endgroup$– Benjamin SteinbergCommented Oct 16, 2021 at 18:31
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$\begingroup$ @BenjaminSteinberg Yes, I also consider groups $G$ which are not finitely generated, such as quotient groups of $\mathbb{C}$ (complex numbers). $\endgroup$– HuiRongCommented Oct 18, 2021 at 0:28
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