TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27

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The matrix category over a ring is additive

Statement

For every ring $R$ , the matrix category ${M a t}_{R}$ is additive.

Facts & Assumptions

Given: A ring $R$ and its matrix category ${M a t}_{R}$ .

[L1]

An additive category is a preadditive category with finite biproducts (Additive category).

[L2]

In ${M a t}_{R}$ , objects are natural numbers and morphisms $n \to m$ are $m \times n$ matrices, with composition by matrix multiplication and identities $I_{n}$ (The matrix category over a ring).

Proof

technique · direct

1.1

For fixed objects $n, m$ , the hom-set ${M a t}_{R} (n, m)$ is the set of $m \times n$ matrices. Entrywise addition makes it an abelian group because $R$ is an abelian group under addition, and matrix multiplication is bilinear with respect to entrywise addition. So ${M a t}_{R}$ is preadditive.

1.2

The object $0$ is a zero object by [L2]. For $m, n$ , the object $m + n$ carries the usual block projections and injections. Given matrices into or out of $m + n$ , the product and coproduct universal properties are exactly the familiar pairing and copairing of block columns and block rows. Thus $m + n$ is a biproduct of $m$ and $n$ , and iterating gives all finite biproducts.

L2construct

2.1

Hence ${M a t}_{R}$ is preadditive and has all finite biproducts, so it is additive by [L1].

L1step 1.1step 1.2∎

Depends on

Used by

The matrix category is fully faithful in modules and, with chosen bases, equivalent to finite free modules Theorem

Dependency tree · two levels

7 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, Proposition 26 (standard reference, not scraped)