False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27

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FALSE: every idempotent splits

Statement

False claim: every idempotent in every preadditive category splits.

Facts & Assumptions

Given: The one-object preadditive category attached to the commutative ring $R = Z \times Z$ and the idempotent $e = (1, 0) \in R$ .

[L1]

Idempotent completeness means that every idempotent splits (Idempotent complete category).

[L2]

An additive category with kernels is idempotent complete, so the missing hypothesis really is extra structure (An additive category with kernels is idempotent complete).

[L3]

The Karoubi envelope adjoins formal splitting objects for idempotents (The idempotent completion of a preadditive category).

Refutation

technique · direct

1.1

The element $e = (1, 0)$ satisfies $e^{2} = e$ , so it is an idempotent endomorphism of the unique object of the one-object preadditive category attached to $R$ .

givenL1

2.1

If $e$ split there as $i p = e$ and $p i = 1$ , then commutativity of $R$ would give $e = i p = p i = 1$ , contradicting $e = (1, 0) \neq (1, 1)$ . So this idempotent does not split.

step 1.1

3.1

Therefore not every idempotent splits. The point of [L2] is that kernels rule this out in additive categories, while [L3] records the standard repair that adjoins the missing split object.

L2L3step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Dixy Msapato, The Karoubi envelope and weak idempotent completion of an extriangulated category, Section 2.1 (standard reference, not scraped)