# Answer to Question #16818 in Algebra for sanches

Question #16818

Let I be a right ideal in a polynomial ring A = R[x1, . . . , xn], where R is any ring. Assume R is a commutative ring or a reduced ring, and let f ∈ A. If f • g = 0 for some nonzero g ∈ A, show that f • r = 0 for some nonzero r ∈ R.

Expert's answer

If

*R*is commutative, we have (*fA*)*g*=*fgA*= 0, so if*I · g*= 0 for some nonzero*g**∈**A*, then*I · r*= 0 for some nonzero*r**∈**R*. If*R*is reduced, then*A*is also reduced. Thus (*gf*)^{2}=*g*(*fg*)*f*= 0 implies that*gf*= 0, and hence (*fAg*)^{2}= 0. It follows that (*fA*)*g*= 0, so if*I · g*= 0 for some nonzero*g**∈**A*, then*I · r*= 0 for some nonzero*r**∈**R*again.Need a fast expert's response?

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