DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-03

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Irreducible and prime elements of an integral domain

Definition

Let $R$ be an integral domain and let $p \in R$ be nonzero and not a unit. The element $p$ is irreducible if every factorisation $p = a b$ has $a$ or $b$ a unit. It is prime if

$p ∣ a b ⟹ p ∣ a or p ∣ b$

for all $a, b \in R$ .

Depends on

Dependency tree · 2 levels

Sources

Sharifi, Abstract Algebra, Advanced Ring Theory