Let
$K$
be a commutative ring with unity,
$R$
an associative
$K$
-algebra of characteristic different from
$2$
with unity element and no nonzero nil right ideal, and
$f({x}_{1} , \ldots , {x}_{n} )$
a multilinear polynomial over
$K$
. Assume that, for all
$x\in R$
and for all
${r}_{1} , \ldots , {r}_{n} \in R$
there exist integers
$m= m(x, {r}_{1} , \ldots , {r}_{n} )\geq 1$
and
$k= k(x, {r}_{1} , \ldots , {r}_{n} )\geq 1$
such that
$\mathop{[{x}^{m} , f({r}_{1} , \ldots , {r}_{n} )] }\nolimits_{k} = 0$
. We prove that: (1) if
$\text{char} (R)= 0$
then
$f({x}_{1} , \ldots , {x}_{n} )$
is central-valued on
$R$
; and (2) if
$\text{char} (R)= p\gt 2$
and
$f({x}_{1} , \ldots , {x}_{n} )$
is not a polynomial identity in
$p\times p$
matrices of characteristic
$p$
, then
$R$
satisfies
${s}_{n+ 2} ({x}_{1} , \ldots , {x}_{n+ 2} )$
and for any
${r}_{1} , \ldots , {r}_{n} \in R$
there exists
$t= t({r}_{1} , \ldots , {r}_{n} )\geq 1$
such that
${f}^{{p}^{t} } ({r}_{1} , \ldots , {r}_{n} )\in Z(R)$
, the center of
$R$
.