Associative rings and algebras Definition

Goldie ring

Builds on Ring Annihilator Ascending chain condition Uniform module Noetherian ring

Definition

Let $R$ be a ring with identity. For a subset $X \subseteq R$ , the right annihilator of $X$ is $r(X) = \set{a \in R : x a = 0 \text{ for all } x \in X}$ , a right ideal of $R$ . The ring $R$ is right Goldie if:

$R$ has finite right uniform dimension: there is no infinite family of nonzero right ideals $I_1, I_2, \ldots$ of $R$ whose sum is direct;
$R$ satisfies the ascending chain condition on right annihilators: every increasing chain $r(X_1) \subseteq r(X_2) \subseteq \cdots$ is eventually constant.

Left Goldie rings are defined in the same way with left ideals and left annihilators. A ring is Goldie if it is both right and left Goldie.

Remarks

Example: a right Noetherian ring $R$ is right Goldie. An infinite direct sum $I_1 \oplus I_2 \oplus \cdots$ of nonzero right ideals would give a strictly increasing chain $I_1 \subset I_1 \oplus I_2 \subset \cdots$ , which is impossible. Right annihilators are right ideals, so they satisfy the ascending chain condition.

Example: every commutative integral domain is Goldie, since two nonzero ideals $I, J$ always meet in the nonzero ideal $IJ$ , and the only annihilators are $0$ and $R$ .

Non-example: let $k$ be a field and $R = \prod_{n \ge 1} k$ . The ideals $I_n = k e_n$ , where $e_n$ has $1$ in coordinate $n$ and $0$ elsewhere, are nonzero and their sum is direct, so clause 1 fails.

Clause 1 can be stated with uniform modules: $R_R$ has finite uniform dimension exactly when it contains an essential submodule that is a finite direct sum of uniform right ideals.

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Annihilator concept
Ascending chain condition concept
Noetherian ring concept
Ring concept
Uniform module concept
Goldie ring definition

Tag	`4AY3RMCTA3`
URI	`m2://4AY3RMCTA3`
Kind	definition
Writing	W0 stub
MSC	16P60, 16U20
Defines	Goldie ring
Uses	Ring, Annihilator, Ascending chain condition, Uniform module, Noetherian ring

Goldie ring

Definition

Remarks

Reading path

Further reading

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