For the special case of ideals it states that every ideal of a Noetherian ring is a finite intersection of primary ideals.
32.
In general, a Noetherian ring is called a Cohen Macaulay ring if the localizations at all maximal ideals are Cohen Macaulay.
33.
For example, a ring in which there is no strictly increasing infinite chain of left ideals is called a left Noetherian ring.
34.
Noether gave an example of a non-commutative Noetherian ring with a right ideal that is not an intersection of primary ideals.
35.
For example, finite direct sums of Noetherian rings are Noetherian, as is the ring of formal power series over a Noetherian ring.
36.
For example, finite direct sums of Noetherian rings are Noetherian, as is the ring of formal power series over a Noetherian ring.
37.
For a Noetherian ring " R ", finitely generated, finitely presented, and coherent are equivalent conditions on a module.
38.
The Lasker Noether theorem for modules states every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules.
39.
Since k [ x _ 1, \ ldots, x _ n ] is a Noetherian ring, there exists an integer m such that
40.
In particular, \ operatorname { Spec } A is a noetherian scheme if and only if " A " is a noetherian ring.
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