DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-02

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The sum $I+J$ and product $IJ$ of two-sided ideals

Definition

The sum $I+J$ and product $IJ$ of two-sided ideals.

For two-sided ideals $I,J\mathrel{\trianglelefteq}R$ , set

$I+J:=\{i+j:i\in I,\ j\in J\},\qquad IJ:=\left\{\sum_{k=1}^m i_kj_k:m\ge0,\ i_k\in I,\ j_k\in J\right\}.$

The empty sum in the definition of $IJ$ is $0$ .

Depends on

Left, right and two-sided ideals

Used by

The sum and product of two-sided ideals are two-sided ideals Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 10 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Janssen and Lindsey, Rings with Inquiry, Ideals (standard reference, not scraped)

The sum I+JI+JI+J and product IJIJIJ of two-sided ideals