On PI-subrings of matrix rings over some classes of skew fields (Q1109855)

Let \(\Delta\) be either the (skew) field of fractions of a group ring KG, where G is a torsion free nilpotent group, or the universal field of fractions of a free group ring, or the universal field of fractions of a group ring of a free soluble group, or else the universal enveloping algebra of a finite-dimensional Lie algebra over a field of characteristic zero. It is proved that if \(D_{k\times k}\) is a matrix subring of \(\Delta_{n\times n}\) where D is a field of dimension \(m^ 2\) over its center, then the PI-degree of \(D_{k\times k}\) divides n (see Corollary 1.3). It is a generalization of Schofield's (1985) result which states that if E is a subfield of \(\Delta_{n\times n}\) where \(\Delta\) is the universal field of fractions of a free group ring, then the PI-degree of E divides n.