CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-17

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Finite direct sums preserve and reflect Noetherian and Artinian conditions

Statement

A finite direct sum is Noetherian if and only if every summand is Noetherian, and it is Artinian if and only if every summand is Artinian. The empty direct sum is included. See Noetherian and Artinian conditions are each exact in short exact sequences.

Facts & Assumptions

Given: The hypotheses and objects in the Statement.

[L1]

In a short exact sequence $0 \to N \to M \to Q \to 0$ , the module $M$ is Noetherian if and only if $N$ and $Q$ are Noetherian; the same equivalence holds with “Artinian” in place of “Noetherian”. (Noetherian and Artinian conditions are each exact in short exact sequences).

[L2]

Let $R$ be a unital ring and $(M_{i})_{i \in I}$ a family of left $R$ -modules (def-left-and-right-modules). 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 $⨁_{i \in I} M_{i} = {m \in \prod_{i \in I} M_{i} : supp (m) is finite}$ (def-submodule). (The direct sum of an indexed family of modules).

Proof

technique · direct

1.1

Induct on the number of summands using the split short exact sequence for a binary direct sum.

L1L2givenalgebra

2.1

Reflection follows because every summand is a submodule and a quotient.

step 1.1givenalgebra

3.1

Both extremes of the induction are admitted. The empty direct sum is the zero module, whose only chains of submodules are constant, so it satisfies both conditions while the "every summand" side is vacuously true; a single summand makes the direct sum that summand, so the equivalence is an identity and supplies the base of the induction in step 1.1. This proves the stated claim.

step 1.1step 2.1givenalgebra∎

Depends on

Used by

Finitely generated modules over a left Noetherian ring are Noetherian Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 22 results over 11 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

William Crawley-Boevey, Noncommutative Algebra, Chapter 1 Sections 1.1-1.9 (standard reference, not scraped)