Infinitesimally PI radical algebras

The Golod-Shafarevich examples show that not every finitely generated nil algebraA is nilpotent. On the other hand, Kaplansky proved that every finitely generated nil PI-algebra is indeed nilpotent. We generalise Kaplansky’s result to include those algebras that are only infinitesimally PI. An associative algebraA is infinitesimally PI whenever the Lie subalgebra generated by the first homogeneous component of its graded algebra gr(A)=⊕ _t⩾1 A ⁱ /A ⁱ⁺¹ is a PI-algebra. We apply our results to a problem of Kaplansky’s concerning modular group algebras with radical augmentation ideal. The author is supported by NSERC of Canada.