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.
If is a unit, then
Statement
If is a commutative ring, , , and is a unit, then
Facts & Assumptions
Given: as in the statement.
The adjugate identity is (For every positive-sized square matrix over a commutative ring, ).
A positive-sized square matrix whose determinant is a unit is invertible (A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit).
A two-sided inverse in a monoid is unique (In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided).
Proof
Put . Scaling [L1] by gives .
By [L2], exists. Both it and are two-sided inverses of , so [L3] makes them equal.
Depends on
- For every positive-sized square matrix over a commutative ring, $A\operatorname{adj}(A)=\operatorname{adj}(A)A=\det(A)I$
- A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit
- In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided
Used by
- Multiplication by 2 on ℤ is injective but not surjective: its determinant is the non-unit 2, its adjugate is integral, and its inverse exists after extending scalars to ℚ Example
- The adjugate gives the inverse of a 3×3 rational matrix with determinant 3 Example
- Cramer's rule over a commutative ring: every solution satisfies det(A)xⱼ=det(Aⱼ(b)), and a unit determinant gives the unique quotient formula Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 31 results over 17 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
- András Pál, Introduction to Commutative Algebra, Proposition 7.2 (standard reference, not scraped)