• A NOTE ON COMMUTATIVITY OF PERIODIC NEAR-FIELD SPACES OVER NEAR-FIELDS

Abstract

A near-field space N satisfying a polynomial identity of the form ab = f (a, b), where f (A, B) is different word from that AB, must have nil Commutator sub near-field space. First major theorem extends this result to the case where f (A, B) varies with a, b with the restriction that all f (A, B) have length at least three and are not of the form AⁿB or ABⁿ. Further restrictions on the f (A, B) are then shown to yields commutativity of a near-field space N. Among these a semi simple sub near-field space and a near-field space specifically that each f (A, B) begins with B and has at least 2 in A. The final theorem establishes commutativity of near-field spaces N satisfying ab = bas where s = s(a, b) is an element of the center of the sub near-field space generated by a and b. All near-field spaces considered are either periodic by hypothesis or turn out to be periodic near-field spaces in the course of the in depth study and investigation of the near-field spaces.