Central zero divisors in group algebras

Smith, Martha K.

doi:10.1090/S0002-9939-1984-0746082-X

Central zero divisors in group algebras
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Proc. Amer. Math. Soc. 91 (1984), 529-531 Request permission

Abstract:

A central element of the complex group algebra ${\mathbf {C}}G$ which is a zero divisor in the ${W^ * }$ group algebra $W(G)$ is also a zero divisor in ${\mathbf {C}}G$. As a corollary, if $K$ is a field of characteristic zero, $G$ is a group, $A$ is an abelian normal subgroup of $G$, and $R$ is the Ore localization of $KG$ obtained by inverting all nonzero elements of $KA$, then all matrix rings over $R$ are directly finite and $R$ has the invariant basis property.

References

K. R. Goodearl, Localized group rings, the invariant basis property and Euler characteristics, Proc. Amer. Math. Soc. 91 (1984), no. 4, 523–528. MR 746081, DOI 10.1090/S0002-9939-1984-0746081-8
Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0470211