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