D. M. Riley and Mark C. Wilson, Communications in Algebra 27 (1999), 3545-3556.
Abstract: Let $latex K$ be a field of characteristic $latex p>0$. Denote by $latex omega(R)$ the augmentation ideal of either a group algebra $latex R=K[G]$ or a restricted enveloping algebra $latex R=u(L)$ over $latex K$. We first characterize those $latex R$ for which $latex omega(R)$ satisfies a polynomial identity not satisfied by the algebra of all $latex 2times 2$ matrices over $latex K$. Then, we examine those $latex R$ for which $latex omega(R)$ satisfies a semigroup identity (that is, a polynomial identity which can be written as the difference of two monomials).