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| Paper IPM / M / 7381 |
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| Abstract: | |
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Let R be a ring and f(x1,…, xn) be a polynomial in
noncommutative indeterminates x1, …,xn with integral
coefficients and zero constant. The ring R is said to be an
f-ring if f(r1,…,rn)=0 for all r1,…,rn of
R and a virtually f-ring if for every n infinite subsets
X1,…,Xn (not necessarily distinct) of R, there exist
n elements r1 ∈ X1,…,rn ∈ Xn such that
f(r1,…,rn)=0. Let ―f be the image of f in
Z[x1,…,xn] (the ring of polynomials with coefficients in
Z in commutative indeterminates x1,…,xn). In this
paper, we show that if ―f ≠ 0, then every left primitive
virtually f-ring is finite. As applications, we prove that if
―f ≠ 0, then every infinite semisimple virtually f-ring
is a commutative f-ring and also if f(x)=∑ni=2aixi+ϵx ∈ Z[x], where ϵ ∈ {−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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