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. 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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