Some Properties on Rings with Units Satisfying a Group Identity

For convenience, a ring with units satisfying a group identity will be called a GI-ring. We show that GI-rings have the following properties which are also properites of PI-rings. (1) Any GI-ring is Dedekind finite (von Neumann finite). (2) Nilpotent elements of a semiprimitive GI-ring have bounded index. (3) The Kurosh problem has a positive answer for GI-algebras, namely, any algebraic GI-algebra is locally finite. We also study Hartleyʹs problem for algebraic GI-algebras.