If is a noncommutative ring, then Morita theory tells us that cannot in general be recovered from its category of modules; that is, there can be a ring , not isomorphic to , such that . This means, for example, that “free” is not a categorical property of modules, since it depends on a choice of ring , or equivalently on a choice of forgetful functor.

It’s therefore something of a surprise that “finitely presented” *is* a categorical property of modules, and hence that it does not depend on a choice of ring . The reason is that being finitely presented is equivalent to a categorical property called compactness.

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