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The matrix category is the finite biproduct completion of a ring
Theorem A one-object preadditive category is the same thing as a ring lets us read a ring as a one-object preadditive category. Definition The matrix category over a ring then adjoins formal finite direct sums of that one object, recorded as the natural numbers, and theorem The matrix category is fully faithful in modules and, with chosen bases, equivalent to finite free modules identifies the result with the finitely generated free -modules. In that sense, is the finite-biproduct completion of .
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Sources
- Kiran S. Kedlaya, Solid modules over an ordinary ring, Examples 1.2.2 and 1.2.6 (standard reference, not scraped)
- Gabriele Lobbia, Wojciech Rozowski, Ralph Sarkis, and Fabio Zanasi, Quantitative Monoidal Algebra, Definition 25 (standard reference, not scraped)