The direct sum of an indexed family of modules

Depends on

• Unital left and right modules over a ring; unqualified module means left module
• Submodule of a module

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
• Finite direct sums preserve and reflect Noetherian and Artinian conditions Corollary
• Localisation commutes with finite intersections of submodules Corollary
• An infinite coproduct need not agree with the infinite product Counterexample
• 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
• Tensor algebra of a vector space Definition
• Tensor product of abelian sheaves and its total complex Definition
• The free module on a set and its standard basis Definition
• 0→ A→ A⊕ C→ C→0 is canonically split Example
• A module-valued coend computed as a quotient of a direct sum Example
• Localised Hom can fail without finite presentation of the source 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
• Associator, symmetry and unitors of the abelian sheaf tensor product Lemma
• Derived tensor product of abelian sheaves Lemma
• Every abelian group embeds in a divisible abelian group Lemma
• For a commutative ring, Hom_R(Rⁿ,N)≅ Nⁿ Lemma
• Graded modules with degree-zero maps form an abelian category Lemma
• Submodules of finite modules over a Noetherian ring are finite by induction Lemma
• The kernel of a finite direct sum is the intersection of the kernels Lemma
• A left transversal identifies Ind_H^G W with a direct sum of [G:H] copies of W Proposition
• Direct-sum, dual, Hom, and tensor representations Proposition
• A module-valued coend is the direct sum of the diagonal values modulo the dinaturality submodule Theorem
• Finite-dimensional kG-modules decompose as finite direct sums of indecomposables uniquely up to order and isomorphism Theorem
• Products of injective modules are injective, with the exact choice boundary Theorem
• Tensor products commute with arbitrary direct sums Theorem
• Universal property of a direct sum of modules Theorem

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)