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.