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Let $R$ be a ring and $f(x_1,\cdots , x_n)$ be a polynomial in
noncommutative indeterminates $x_1, \cdots ,x_n$ with integral
coefficients and zero constant. The ring $R$ is said to be an
$f$-ring if $f(r_1,\cdots ,r_n)=0$ for all $r_1,\cdots ,r_n$ of
$R$ and a {\it virtually f-ring} if for every $n$ infinite subsets
$X_1,\cdots ,X_n$ (not necessarily distinct) of $R$, there exist
$n$ elements $r_1\in X_1,\cdots ,r_n\in X_n$ such that
$f(r_1,\cdots ,r_n)=0.$ Let $\bar{f}$ be the image of $f$ in
$Z[x_1,\cdots ,x_n]$ (the ring of polynomials with coefficients in
$Z$ in commutative indeterminates $x_1,\cdots ,x_n)$. In this
paper, we show that if $\bar{f}\neq 0$, then every left primitive
virtually $f$-ring is finite. As applications, we prove that if
$\bar{f}\neq 0$, then every infinite semisimple virtually $f$-ring
is a commutative $f$-ring and also if $f(x)=\sum^n_{i=2}
a_ix^i+\epsilon x\in Z[x],$ where $\epsilon \in\{-1,1\}$, then
every infinite virtually $f$-ring is a commutative $f$-ring.
Finally we show that every commutative Noetherian virtually
$f$-ring with identity is finite.
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