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Paper   IPM / M / 7381
School of Mathematics
  Title:   Rings virtually satisfying a polynomial identity
  Author(s):  S. Akbari (Joint with A. Abdollahi)
  Status:   Published
  Journal: J. Pure Appl. Algebra
  Vol.:  198
  Year:  2005
  Pages:   9-19
  Supported by:  IPM
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 r1X1,…,rnXn 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=2aixixZ[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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