11. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. 12. For example, every quasi-projective scheme over a Noetherian ring has the resolution property. 13. For commutative Noetherian rings , this is the same as the definition using chains of prime ideals. 14. That follows because the rings of algebraic geometry, in the classical sense, are Noetherian rings . 15. Both facts imply that a finitely generated commutative algebra over a Noetherian ring is again a Noetherian ring. 16. Both facts imply that a finitely generated commutative algebra over a Noetherian ring is again a Noetherian ring . 17. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension. 18. All quotient rings of a Noetherian ring are Noetherian, but that does not necessarily hold for its subrings. 19. Over a commutative Noetherian ring , this gives a particularly nice understanding of all injective modules, described in. 20. A "'Gorenstein ring "'is a commutative Noetherian ring such that each Cohen Macaulay.
