Hilbert basis theorem
This article gives the statement, and possibly proof, of a commutative unital ring property satisfying a commutative unital ring metaproperty
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Statement
Property-theoretic statement
The property of commutative unital rings of being Noetherian is polynomial-closed.
Verbal statement
The polynomial ring over a Noetherian ring is Noetherian.
Symbolic statement
Let be a Noetherian ring. Then the polynomial ring (where is an indeterminate) is also a Noetherian ring.
![{\displaystyle A[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7d5d3b3ae4cfb46d3ef305e5d7dcb130f10e468)
Proof
The proof rests on the notion of the leading coefficient map and the fact that if the image of an ideal under the leading coefficient map is finitely generated, so is the original ideal.
Further information: image under leading coefficient map is finitely generated implies finitely generated