# Non zerodivisors in ideals of polynomial rings [duplicate]

The question is the following:

Let $$f$$ be a polynomial of the ring $$R[x_1, \ldots, x_n]$$, with $$R$$ any ring, and let $$\mathrm{cont}(f)$$ be the ideal generated by the coefficients of $$f$$. Why if $$\mathrm{cont}(f)$$ contains a non-zerodivisor of $$R$$, then $$f$$ is a non-zerodivisor of $$R[x_1, \ldots, x_n]$$?

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Hint: When you write that non zero divisor $$a$$ as a linear comnbination, this gives you a recipe for a polynomial $$g$$ such that one coefficient of $$f\cdot g$$ is $$a$$. Now assume $$h\cdot f=0$$. Then also $$hfg=0$$.