Invertible and nilpotent elements in the group algebra of a unique product group.
Let \(k\) be a commutative ring with unity, \(G\) a group and \(kG\) its group algebra. Kaplansky's problem asks whether the group ring of a torsion-free group over a zero-divisor free ring is also zero-divisor free. There is a well-known result which states that the crossed product of a u.p. group (which is torsion-free) over a zero-divisor free ring is also zero-divisor free and has trivial units only. The author applies tools of algebraic geometry to describe the upper nil radical (group of units) of \(kG\) in case \(KG\) is zero-divisor free (has trivial units only) for every field \(K\) which is a \(k\)-algebra, conditions satisfied for a u.p. group. It is proved that the upper nil radical is the group algebra of \(G\) over the nil radical of \(k\), which coincides with the set of all nilpotent elements, and units decompose into the sum of trivial units according to a direct sum decomposition of the ring \(k\) plus a nilpotent element.
scientific article; zbMATH DE number 5644660