Artin–Tate lemma

In algebra, the Artin–Tate lemma, named after Emil Artin and John Tate, states:

Let A be a commutative Noetherian ring and ${\displaystyle B\subset C}$ commutative algebras over A. If C is of finite type over A and if C is finite over B, then B is of finite type over A.

(Here, "of finite type" means "finitely generated algebra" and "finite" means "finitely generated module".) The lemma was introduced by E. Artin and J. Tate in 1951 to give a proof of Hilbert's Nullstellensatz.

The lemma is similar to the Eakin–Nagata theorem, which says: if C is finite over B and C is a Noetherian ring, then B is a Noetherian ring.