Answer to Question #16818 in Algebra for sanches

Question

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.

Answer 1

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.