Group identities on units of rings
Let \(A\) be an algebra over an infinite commutative domain \(R\), and let \(R^{-1} A\) be the ring of fractions with respect to the non-zero elements of \(R\). It is shown that all idempotents of \(R^{-1} A\) are central provided \(A\) is semiprime and its unit group \({\mathcal U}(A)\) satisfies a group identity. As an application we prove, as conjectured by B. Hartley, that the group ring \(RG\) satisfies a polynomial identity provided \({\mathcal U}(RG)\) satisfies a group identity and \(G\) is a \(p'\)- group, where \(p\) is the characteristic of \(R\).
scientific article; zbMATH DE number 687021