# criterion for a module to be noetherian

###### Theorem 1.

A module is noetherian^{} if and only if all of its submodules and
quotients (http://planetmath.org/QuotientModule) are noetherian.

###### Proof.

Suppose $M$ is Noetherian (over a ring $R$), and $N\subseteq M$ a submodule. Since any submodule of $M$ is finitely generated^{}, any submodule of $N$, being a submodule of $M$, is finitely generated as well. Next, if $A/N$ is a submodule of $M/N$, and if ${a}_{1},\mathrm{\dots},{a}_{n}$ is a generating set for $A\subseteq M$, then ${a}_{1}+N,\mathrm{\dots},{a}_{n}+N$ is a generating set for $A/N$. Conversely, if every submodule of $M$ is Noetherian, then $M$, being a submodule itself, must be Noetherian.
∎

A weaker form of the converse is the following:

###### Theorem 2.

If $N\mathrm{\subseteq}M$ is a submodule of $M$ such that $N$ and $M\mathrm{/}N$ are Noetherian, then $M$ is Noetherian.

###### Proof.

Suppose ${A}_{1}\subseteq {A}_{2}\subseteq \mathrm{\cdots}$ is an ascending chain of submodules of $M$. Let ${B}_{i}={A}_{i}\cap N$, then ${B}_{1}\subseteq {B}_{2}\subseteq \mathrm{\cdots}$ is an ascending chain of submodules of $N$. Since $N$ is Noetherian, the chain terminates at, say ${B}_{n}$. Let ${C}_{i}=({A}_{i}+N)/N$, then ${C}_{1}\subseteq {C}_{2}\subseteq \mathrm{\cdots}$ is an ascending chain of submodules of $M/N$. Since $M/N$ is Noetherian, the chain stops at, say ${C}_{m}$. Let $k=\mathrm{max}(m,n)$. Then we have ${B}_{k}={B}_{k+1}$ and ${C}_{k}={C}_{k+1}$. We want to show that ${A}_{k}={A}_{k+1}$. Since ${A}_{k}\subseteq {A}_{k+1}$, we need the other inclusion. Pick $a\in {A}_{k+1}$. Then $a+N=b+N$, where $b\in {A}_{k}$. This means that $a-b\in N$. But $b\in {A}_{k+1}$ as well, so $a-b\in N\cap {A}_{k+1}$. Since $N\cap {A}_{k}=N\cap {A}_{k+1}$, this means that $a-b\in {A}_{k}$ or $a\in {A}_{k}$. ∎

Title | criterion for a module to be noetherian |
---|---|

Canonical name | CriterionForAModuleToBeNoetherian |

Date of creation | 2013-03-22 15:28:46 |

Last modified on | 2013-03-22 15:28:46 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 9 |

Author | mps (409) |

Entry type | Theorem |

Classification | msc 13E05 |