DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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The direct sum of an indexed family of modules

Definition

Let $R$ be a unital ring and $(M_{i})_{i \in I}$ a family of left $R$ -modules (Unital left and right modules over a ring; unqualified module means left module). Their direct product is the module $\prod_{i \in I} M_{i}$ with coordinatewise operations. The support of $m = (m_{i})$ is ${i \in I : m_{i} \neq 0}$ , and the direct sum is the submodule $⨁_{i \in I} M_{i} = {m \in \prod_{i \in I} M_{i} : supp (m) is finite}$ (Submodule of a module).

This subset is a submodule because the support of a sum is contained in the union of two finite supports and scalar multiplication cannot enlarge support.

For each $i$ , the coordinate inclusion $ȷ_{i} : M_{i} \to ⨁_{j} M_{j}$ puts its input in coordinate $i$ and zero elsewhere. If $I = \emptyset$ , both product and direct sum are the zero module.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 10 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

A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
The Stacks Project, Algebra (standard reference, not scraped)
P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)