DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

The opposite ring $R^{o p}$

Definition

For a unital ring $R$ , the opposite ring $R^{o p}$ has the same underlying abelian group, identity, and addition as $R$ , with multiplication $a ⋆ b : = b a$ . Associativity and both distributive laws follow from those of $R$ with the order reversed, and the same element $1$ is a two-sided identity. Thus the displayed operations really form a unital ring, including when $R$ is the zero ring.

Depends on

Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides

Used by

Left and right semisimplicity of a ring agree Corollary
End_R(_R R)≅ Rᵒᵖ Theorem
Wedderburn–Artin theorem for semisimple rings Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 10 results over 10 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)

The opposite ring RopRop