# What does Artinian mean?

## What does Artinian mean?

Filters. (algebra, of a ring) In which any descending chain of ideals eventually starts repeating. adjective. (algebra, of a module) In which any descending chain of submodules eventually starts repeating.

## What is maximal ideal of ring?

In the ring Z of integers, the maximal ideals are the principal ideals generated by a prime number. More generally, all nonzero prime ideals are maximal in a principal ideal domain. The ideal is a maximal ideal in ring .

**Are all prime ideals maximal?**

(1) An ideal P in A is prime if and only if A/P is an integral domain. (2) An ideal m in A is maximal if and only if A/ m is a field. Of course it follows from this that every maximal ideal is prime but not every prime ideal is maximal.

### Is a field Artinian?

Then R⊇aR⊇a2R⊇⋯ R ⊇ a R ⊇ a 2 R ⊇ ⋯ . As R is Artinian, there is some n∈N n ∈ ℕ such that anR=an+1R R = a n + 1 ….an Artinian integral domain is a field.

Title | an Artinian integral domain is a field |
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Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 13 |

Author | yark (2760) |

### Why is Z Noetherian?

Any PID is noetherian: any ideal is generated by one element. So Z is noetherian. Note that the ring itself is finitely generated (by the element 1), but there are ideals that are not finitely generated. Remark Let R be a ring such that every prime ideal is finitely generated.

**What is Noetherian R module?**

From Wikipedia, the free encyclopedia. In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion.

## Is Z √ 7 an UFD?

How is Z[√7] a UFD? UFD means unique factorisation domain. And Z[√7] = {a+b√7: a and b are in Z}, Z is a ring of integers.