Alphabeta MathHow 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
Definition
For a unital ring , the opposite ring has the same underlying abelian group, identity, and addition as , with multiplication . Associativity and both distributive laws follow from those of with the order reversed, and the same element is a two-sided identity. Thus the displayed operations really form a unital ring, including when is the zero ring.
Depends on
Used by
- Left and right semisimplicity of a ring agree Corollary
- Enveloping algebra and the bimodule–module dictionary Definition
- Left and right global dimension of a ring Definition
- Semisimple rings have vanishing positive Tor and Ext Proposition
- End_R(_R R)≅ Rᵒᵖ Theorem
- The two-sided bar complex is a projective Aᵉ-resolution Theorem
- Weak global dimension is Tor-detected and left-right symmetric Theorem
- Wedderburn–Artin theorem for semisimple rings Theorem
Dependency tree · two levels
8 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
- William Crawley-Boevey, Noncommutative Algebra, Chapter 1 Sections 1.1-1.9 (standard reference, not scraped)