ON NAKAYAMA'S EXTENSION OF THE x»<*> THEOREMS

where the a,are r fixed nonzero elements of Z, and 0<«i(a) there is an element a in 4 such that xa = 0, ya=y\. Now yan — y\n so that if/(a)£Z is the relation of type (1) that a satisfies, we have xf(a) =0, yf(a) =y/(X). Since/(a) £Z£Z> this makes f(a) =0 so that y/(X) =0, /(X) =0, and by Nakayama's result D=Z. Thus the center of a primitive ring satisfying (1) is a field. Moreover, we have proved that each element in this field satisfies a polynomial equation of the form f(a)=0 with / as in (1). Let P be the prime subfield of Z and Q be the field obtained by adjoining to P a maximal, algebraically independent set from among ai, • • • , ar. Assuming that Z is not absolutely algebraic1 of prime characteristic, Lemmas 1 and 2 of [6] show that Z is purely inseparable over Q or Z = Q. In the former case f(a)p =0 is an equation with coefficients in the rational function field Q; thus in either case every nonzero rational function X would satisfy an equation