equivalent defining conditions on a Noetherian ring

Let $R$ be a ring. Then the following are equivalent:

1.

every left ideal of $R$ is finitely generated,
2.

the ascending chain condition on left ideals holds in $R$ ,
3.

every non-empty family of left ideals has a maximal element.

Proof.

$(1\Rightarrow 2)$ . Let $I_{1}\subseteq I_{2}\subseteq\cdots$ be an ascending chain of left ideals in $R$ . Let $I$ be the union of all $I_{j}$ , $j=1,2,\ldots$ . Then $I$ is a left ideal, and hence finitely generated, by, say, $a_{1},\cdots a_{n}$ . Now each $a_{i}$ belongs to some $I_{\alpha_{i}}$ . Take the largest of these, say $I_{\alpha_{k}}$ . Then $a_{i}\in I_{\alpha_{k}}$ for all $i=1,\ldots,n$ , and therefore $I\subseteq I_{\alpha_{k}}$ . But $I_{\alpha_{k}}\subseteq I$ by the definition of $I$ , the equality follows.

$(2\Rightarrow 3)$ . Let $\mathcal{S}$ be a non-empty family of left ideals in $R$ . Since $\mathcal{S}$ is non-empty, take any left ideal $I_{1}\in\mathcal{S}$ . If $I_{1}$ is maximal, then we are done. If not, $\mathcal{S}-\{I_{1}\}$ must be non-empty, such that pick $I_{2}$ from this collection so that $I_{1}\subseteq I_{2}$ (we can find such $I_{2}$ , for otherwise $I_{1}$ would be maximal). If $I_{2}$ is not maximal, pick $I_{3}$ from $\mathcal{S}-\{I_{1},I_{2}\}$ such that $I_{1}\subseteq I_{2}\subseteq I_{3}$ , and so on. By assumption, this can not go on indefinitely. So for some positive integer $n$ , we have $I_{n}=I_{m}$ for all $m\geq n$ , and $I_{n}$ is our desired maximal element.

$(3\Rightarrow 1)$ . Let $I$ be a left ideal in $R$ . Let $\mathcal{S}$ be the family of all finitely generated ideals of $R$ contained in $I$ . $\mathcal{S}$ is non-empty since $(0)$ is in it. By assumption $\mathcal{S}$ has a maximal element $J$ . If $J\neq I$ , then take an element $a\in I-J$ . Then $\langle J,a\rangle$ is finitely generated and contained in $I$ , so an element of $\mathcal{S}$ , contradicting the maximality of $J$ . Hence $J=I$ , in other words, $I$ is finitely generated. ∎

A ring satisfying any, and hence all three, of the above conditions is defined to be a left Noetherian ring. A right Noetherian ring is similarly defined.

Title	equivalent defining conditions on a Noetherian ring
Canonical name	EquivalentDefiningConditionsOnANoetherianRing
Date of creation	2013-03-22 18:04:27
Last modified on	2013-03-22 18:04:27
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	6
Author	CWoo (3771)
Entry type	Derivation
Classification	msc 16P40