The Hilbert Basis theorem is probably one of the easiest-to-state theorems that I know of in commutative algebra. The last time I posted about it. Hilbert, D. "Über die Theorie der algebraischen Formen." Math. Ann. 36, , Referenced on Wolfram|Alpha: Hilbert Basis Theorem. CITE THIS AS. HILBERT BASIS THEOREM. MATH NOTES. Definition 1. A ring R is called Noetherian if every ideal in R is finitely gener- ated, i.e. is of the form (a1,a2.

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Finally, by coincidence, just yesterday in my graduate course on local fields I got to the proof of hilbert basis theorem "tensor product theorem" on the classification of norms in a finite-dimensional field extension which came up in a previous MO answer.

And sometimes they are not. For groups they were set-functions which preserved the group operation for all.

The idea hilbert basis theorem preserving algebraic structure translates to rings, but as rings have two operations we must preserve both.

Let be a function on the underlying sets. We call a ring homomorphism if is a group homomorphism of the underlying abelian groups, and for all we have.

In fact, because we can decompose any integer into primes, the integers always stop in this way. This makes the ring of integers noetherian. Notice hilbert basis theorem the integers do not have the descending chain condition; namely, can go on forever without repeating the same term twice.

This does not affect the fact that it is noetherian, as to hilbert basis theorem noetherian we only require the Ascending Chain Condition be satisfied.

We denote it by [math]R[X].

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