Eakin–Nagata theorem

In abstract algebra, the Eakin–Nagata theorem states: given commutative rings ${\displaystyle A\subset B}$ such that ${\displaystyle B}$ is finitely generated as a module over ${\displaystyle A}$, if ${\displaystyle B}$ is a Noetherian ring, then ${\displaystyle A}$ is a Noetherian ring. (Note the converse is also true and is easier.)

The theorem is similar to the Artin–Tate lemma, which says that the same statement holds with "Noetherian" replaced by "finitely generated algebra" (assuming the base ring is a Noetherian ring).

The theorem was first proved in Paul M. Eakin's thesis (Eakin 1968) and later independently by Masayoshi Nagata (1968). The theorem can also be deduced from the characterization of a Noetherian ring in terms of injective modules, as done for example by David Eisenbud in (Eisenbud 1970); this approach is useful for a generalization to non-commutative rings.