Nil and power-central polynomials in rings

Abstract

A polynomial in noncommuting variables is vanishing, nil or central in a ring, R R , if its value under every substitution from R R is 0, nilpotent or a central element of R R , respectively. THEOREM. If R R has no nonvanishing multilinear nil polynomials then neither has the matrix ring R n {R_n} . THEOREM. Let R R be a ring satisfying a polynomial identity modulo its nil radical N N , and let f f be a multilinear polynomial. If f f is nil in R R then f f is vanishing in R / N R/N . Applied to the polynomial x y − y x xy - yx , this establishes the validity of a conjecture of Herstein’s, in the presence of polynomial identity. THEOREM. Let m m be a positive integer and let F F be a field containing no m m th roots of unity other than 1. If f f is a multilinear polynomial such that for some n > 2 f m n > 2{f^m} is central in F n {F_n} , then f f is central in F n {F_n} . This is related to the (non)existence of noncrossed products among p 2 {p^2} -dimensional central division rings.