noetherian ring in a sentence

Examples

1. For the special case of ideals it states that every ideal of a Noetherian ring is a finite intersection of primary ideals.
2. In general, a Noetherian ring is called a Cohen Macaulay ring if the localizations at all maximal ideals are Cohen Macaulay.
3. For example, a ring in which there is no strictly increasing infinite chain of left ideals is called a left Noetherian ring.
4. Noether gave an example of a non-commutative Noetherian ring with a right ideal that is not an intersection of primary ideals.
5. For example, finite direct sums of Noetherian rings are Noetherian, as is the ring of formal power series over a Noetherian ring.
6. For example, finite direct sums of Noetherian rings are Noetherian, as is the ring of formal power series over a Noetherian ring.
7. For a Noetherian ring " R ", finitely generated, finitely presented, and coherent are equivalent conditions on a module.
8. 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.
9. Since k [ x _ 1, \ ldots, x _ n ] is a Noetherian ring, there exists an integer m such that
10. In particular, \ operatorname { Spec } A is a noetherian scheme if and only if " A " is a noetherian ring.