Transcendental Elements in Continuous Rings

Abstract

In (2), John von Neumann introduced the concept of a continuous ring as a generalization to the infinite limiting case of the total matric algebras over a division ring. Von Neumann sketched a theory of arithmetic for such continuous rings and asserted :(⋆) every continuous ring contains purely transcendental elements c.This means: for every polynomial p(t) = tm + Z1tm-1 + . . . + zm {m ≥ 1) which has all coefficients zi in the centre of , the element p(c) has a reciprocal in , that is, (p(c))-l exists such that p(c).(p(c))-l = (p(c))-l.p(c) = 1.