DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-11

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.

Invertible square matrices and similarity over a commutative ring

Definition

Let $R$ be a commutative ring. A matrix $A \in M_{n} (R)$ is invertible when there is $B \in M_{n} (R)$ with $A B = I_{n} = B A$ . The inverse is unique and is denoted $A^{- 1}$ . The general linear group over $R$ is ${GL}_{n} (R) : = {A \in M_{n} (R) : A is invertible} .$

Matrices $A, B \in M_{n} (R)$ are similar over $R$ when there is $P \in {GL}_{n} (R)$ such that $B = P^{- 1} A P$ . Associativity and the identity laws from Matrix arithmetic over a commutative ring is associative, unital and distributive, and transpose reverses products make these the usual units and conjugacy relation in the matrix monoid.

Depends on

Dependency tree · 2 levels

Sources

S. New, MATH 146 Linear Algebra 1 Lecture Notes, Ch. 4