| 1. | Consequently, an irreducible ideal of a Noetherian ring is primary.
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| 2. | A Noetherian ring does not necessarily have a dualizing module.
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| 3. | The integers, however, form a Noetherian ring which is not Artinian.
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| 4. | The decomposition does not hold in general for non-commutative Noetherian rings.
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| 5. | This is sufficient to guarantee that a right-Noetherian ring is right Goldie.
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| 6. | The Krull intersection theorem says that this cannot happen for a Noetherian ring .)
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| 7. | It has only been proven for special types of Noetherian rings, so far.
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| 8. | It has led to a better understanding of noncommutative rings, especially noncommutative Noetherian rings.
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| 9. | In a right noetherian rings; the result is known as Levitzky's theorem.
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| 10. | It is very hard to construct examples of Noetherian rings that are not universally catenary.
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