Nil and Power-Central Polynomials in Rings

Abstract

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