Group algebras and enveloping algebras with nonmatrix and semigroup identities

Let K be a field of characteristic p > 0. Denote by omega(R) the augmentati on ideal of either a group algebra R = K[G] or a restricted enveloping alge bra R = u(L) over K. We first characterize those R for which omega(R) satis fies a polynomial identity not satisfied by the algebra of all 2 x 2 matric es over K. Then, we examine those R for which omega(R) satisfies a semigrou p identity (that is, a polynomial identity which can be written as the diff erence of two monomials).