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Principal and Prime Ideals

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Let R be a commutative ring with 1. Show that the principal ideal generated by x in the polynomial ring R[x] is a prime ideal if and only if R is an integral domain. Prove that (x) is a maximal ideal if and only if R is a field.

I have a solution for this problem but it starts with:First, note that a polynomial f in R[x] belongs to the ideal (x) if and only if f (0) = 0.

Can someone please clarify this note a bit.

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Solution Summary

The expert examines principals and prime ideals generated by a polynomial ring.

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Proof:
"=>": If f in R[x] belongs to the ideal (x), then f(x) = xg(x) for some g(x) in R[x], then ...

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