DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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

A ring with finitely many ideals of zero intersection whose quotients are Noetherian rings is Noetherian Corollary
Every finite-dimensional module is a direct sum of highest-weight modules Corollary
Exact finite tensor functors have projective right-module kernels Corollary
Finite direct sums preserve and reflect Noetherian and Artinian conditions Corollary
Localisation commutes with finite intersections of submodules Corollary
Top cohomology of projective twists Corollary
A coproduct-preserving left exact module functor is not tensor Counterexample
A projective generator need not be small Counterexample
A right exact module functor without coproduct preservation is not tensor Counterexample
An infinite coproduct need not agree with the infinite product Counterexample
Proper cohomology need not be finite for noncoherent sheaves Counterexample
Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules Definition
Associated graded algebra of a filtered algebra Definition
Direct products and direct sums of Lie algebras Definition
Invariant factors and elementary divisors of a finitely generated module over a PID Definition
Irreducible, completely reducible, and faithful representations Definition
PBW filtration on the enveloping algebra Definition
Semisimple modules as direct sums of simple modules Definition
Small projective generators and progenerators Definition
Tensor algebra of a vector space Definition
Tensor product of abelian sheaves and its total complex Definition
The free FI-module M(W) generated by an Sₘ-module Definition
The free module on a set and its standard basis Definition
The Schur idempotent and the Schur functor for type-A q-Schur algebras Definition
The type-A q-Schur algebra and its Weyl modules Definition
0→ A→ A⊕ C→ C→0 is canonically split Example
A module-valued coend computed as a quotient of a direct sum Example
A nonsplit simple has Hom-pairing diagonal two Example
Compact support and softness on a finite discrete space Example
Localised Hom can fail without finite presentation of the source Example
Natural transformations between tensor composites are governed by bimodule maps Example
The ring (ℤ/2)^ℕ is not Noetherian Example
A chain contraction makes the odd-to-even parity map invertible Lemma
A finite-dimensional algebra separates its split simple modules Lemma
A lifted finite CW equivalence has a contractible group-ring mapping cone Lemma
An additive module functor is cocontinuous exactly when it is right exact and preserves coproducts Lemma
Associator, symmetry and unitors of the abelian sheaf tensor product Lemma
Bounded generation degree bounds surjectivity degree and weight Lemma
Canonical free presentations force the comparison to be an isomorphism Lemma
Compact-support acyclicity of c-soft sheaves Lemma

…and 31 more results.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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)