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Paper IPM / M / 7381  


Abstract:  
Let R be a ring and f(x_{1},…, x_{n}) be a polynomial in
noncommutative indeterminates x_{1}, …,x_{n} with integral
coefficients and zero constant. The ring R is said to be an
fring if f(r_{1},…,r_{n})=0 for all r_{1},…,r_{n} of
R and a virtually fring if for every n infinite subsets
X_{1},…,X_{n} (not necessarily distinct) of R, there exist
n elements r_{1} ∈ X_{1},…,r_{n} ∈ X_{n} such that
f(r_{1},…,r_{n})=0. Let ―f be the image of f in
Z[x_{1},…,x_{n}] (the ring of polynomials with coefficients in
Z in commutative indeterminates x_{1},…,x_{n}). In this
paper, we show that if ―f ≠ 0, then every left primitive
virtually fring is finite. As applications, we prove that if
―f ≠ 0, then every infinite semisimple virtually fring
is a commutative fring and also if f(x)=∑^{n}_{i=2}a_{i}x^{i}+ϵx ∈ Z[x], where ϵ ∈ {−1,1}, then
every infinite virtually fring is a commutative fring.
Finally we show that every commutative Noetherian virtually
fring with identity is finite.
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