DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-08-27

How statement and proof provenance work

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The matrix category over a ring

Definition

For a ring $R$ (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides), the matrix category ${M a t}_{R}$ has natural numbers as objects. A morphism $n \to m$ is an $m \times n$ matrix with entries in $R$ . If $A : n \to m$ and $B : m \to p$ , their composite is the $p \times n$ matrix $(B A)_{i k} : = \sum_{j = 1}^{m} b_{i j} a_{j k} .$ The identity on $n$ is the matrix $I_{n}$ with diagonal entries $1_{R}$ and off-diagonal entries $0_{R}$ . These finite sums are defined in the additive abelian group of Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides; associativity and the identity laws follow by finite reindexing together with associativity and distributivity in $R$ .

The zero object is $0$ , since there is exactly one $m \times 0$ and one $0 \times m$ matrix for each $m$ .

Depends on

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

Used by

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

Gabriele Lobbia, Wojciech Rozowski, Ralph Sarkis, and Fabio Zanasi, Quantitative Monoidal Algebra, Definition 25 (standard reference, not scraped)