Some properties on rings with units satisfying a group identity
A ring \(R\) with identity element 1 and group of units \(U(R)\) satisfies a group identity, or is a GI-ring, if for some nontrivial word \(\omega(x_1,\dots,x_n)\) in the free group \(\langle x_1,\dots,x_n,\dots\rangle\) and all \(u_j\in U(R)\), \(\omega(u_1,\dots,u_n)=1\). The author proves that GI-rings satisfy some of the same properties as rings satisfying a polynomial identity (PI-rings). The main results assume that \(R\) is a GI-ring and show: if \(a,b\in R\) with \(ab=1\) then \(ba=1\); if \(R\) is a semiprime algebra over an infinite field then \(R\) has bounded index of nilpotence; and if \(R\) is an algebra which is algebraic over the field \(K\) then \(R\) is locally finite. In the last case, if \(K\) is infinite or if the sum \(N(R)\) of all nilpotent ideals of \(R\) is nilpotent then \(R\) is a PI-ring.
scientific article; zbMATH DE number 1539384