21. Most of them coincide with the Krull dimension for Noetherian rings , but can differ for non-Noetherian rings. 22. Most of them coincide with the Krull dimension for Noetherian rings, but can differ for non-Noetherian rings . 23. Over a Noetherian ring , every injective module is the direct sum of ( uniquely determined ) indecomposable injective modules. 24. I would also endorse keeping " Noetherian rings "-how many people have a mathematical concept named after them? 25. Therefore, for noncommutative Noetherian rings , these two versions coincide and one is justified in talking about the global dimension. 26. In fact, this can be generalized to right noetherian rings ; this result is known as Levitzky's theorem. 27. Since PID's are Noetherian rings , this can be seen as a manifestation of the Lasker-Noether theorem. 28. All Dedekind domains of characteristic 0 and all local Noetherian rings of dimension at most 1 are J-2 rings. 29. By Hilbert's basis theorem and some elementary properties of Noetherian rings , every affine or projective coordinate ring is Noetherian. 30. A ring is left Noetherian if and only if all its left ideals are finitely generated; analogously for right Noetherian rings .
