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.

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