In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian.
If is a ring, let denote the ring of polynomials in the indeterminate over . Hilbert proved that if is "not too large", in the sense that if is Noetherian, the same must be true for . Formally,
Hilbert's Basis Theorem. If is a Noetherian ring, then is a Noetherian ring.
Corollary. If is a Noetherian ring, then is a Noetherian ring.
This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the set of common roots of finitely many polynomial equations. Hilbert proved the theorem (for the special case of polynomial rings over a field) in the course of his proof of finite generation of rings of invariants.
Hilbert produced an innovative proof by contradiction using mathematical induction; his method does not give an algorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist. One can determine basis polynomials using the method of Gröbner bases.
Theorem. If is a left (resp. right) Noetherian ring, then the polynomial ring is also a left (resp. right) Noetherian ring.
Remark. We will give two proofs, in both only the "left" case is considered; the proof for the right case is similar.
Suppose is a non-finitely generated left ideal. Then by recursion (using the axiom of dependent choice) there is a sequence of polynomials such that if is the left ideal generated by then is of minimal degree. It is clear that is a non-decreasing sequence of natural numbers. Let be the leading coefficient of and let be the left ideal in generated by . Since is Noetherian the chain of ideals
must terminate. Thus for some integer . So in particular,
Now consider
whose leading term is equal to that of ; moreover, . However, , which means that has degree less than , contradicting the minimality.
Let be a left ideal. Let be the set of leading coefficients of members of .
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In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian. If is a ring, let denote the ring of polynomials in the indeterminate over . Hilbert proved that if is "not too large", in the sense that if is Noetherian, the same must be true for . Formally, Hilbert's Basis Theorem. If is a Noetherian ring, then is a Noetherian ring. Corollary. If is a Noetherian ring, then is a Noetherian ring.
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