Integral domain

The polynomial ring over an integral domain is an integral domain

Let be a ring and be the polynomial ring in indeterminate . Then is an integral domain iff is an integral domain. #m/thm/ring

Proof

Assume is an integral domain. Clearly is commutative since is. Let be nonzero with leading terms and respectively. Then the leading term of is so .

Note if is an integral domain, then so are its subrings, including .

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