On the Prime Radical of $PI$-Representable Groups

The notion of $PI$-representable groups is introduced; these are subgroups of invertible elements of a $PI$-algebra over a field. It is shown that a $PI$-representable group has a largest locally solvable normal subgroup, and this subgroup coincides with the prime radical of the group. The prime radical of a finitely generated $PI$-representable group is solvable. The class of $PI$-representable groups is a generalization of the class of linear groups because in the groups of the former class the largest locally solvable normal subgroup can be not solvable.