On Nakayama’s extension of the 𝑥^{𝑛(𝑥)} theorems

Abstract

where the oi are r fixed nonzero elements of Z, and 0 <ni(a) <n,(a) (i=2, * * *, r), then A =Z. In [3, Theorem 11; 5; 2; 1] specialized forms of (1) (e.g. an(a) Cz, an(a) -aGZ) are shown to imply commutativity at least for semi-simple rings. It is natural therefore to seek an extension of Nakayama's result to semi-simple rings. Since a semi-simple ring is a subdirect sum of primitive rings and (1) is preserved under homomorphism we first study primitive rings satisfying (1). If A is such a ring it may be identified with a dense ring of linear transformations on a vector space 9 over a division ring D. Since D is the ring of endomorphisms of 91 that commute with A, Z is in D and in fact in the center of D. We may assume that there are at least two independent vectors x, y in W. If XED there is an element a in A such that xa=0, ya=yX. Now yan=yXn so that if f(a)EZ is the relation of type (1) that a satisfies, we have xf(a)= 0, yf(a) =yf(X). Since f(a) E ZCD this makes f(a) = 0 so that yf(X) =0, f(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 f 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 a,i, * * * O r. Assuming that Z is not absolutely algebraic' 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