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# 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.

## Mathematics Subject Classification

16P40*no label found*

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