Editing Ideal theory (section)

== Ideals in a finitely generated algebra over a field ==
{{expand section|date=May 2022}}
{{see also|finitely generated algebra}}

Ideals in a finitely generated algebra over a field (that is, a quotient of a [[polynomial ring]] over a field) behave somehow nicer than those in a general commutative ring. First, in contrast to the general case, if <math>A</math> is a finitely generated algebra over a field, then the [[radical of an ideal]] in <math>A</math> is the intersection of all maximal ideals containing the ideal (because <math>A</math> is a [[Jacobson ring]]). This may be thought of as an extension of [[Hilbert's Nullstellensatz]], which concerns the case when <math>A</math> is a polynomial ring.<!--On the other hand, the [[Noether normalization lemma]]-->