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Theorem 17.20

[theorem] For $R = M_n(S)$ (for any fixed $n \ge 1$), the module categories $\operatorname{Mod}{-}R$ and $\operatorname{Mod}{-}S$ are additively equivalent. [/theorem]

Remarks 17.23

The (additive) categorical equivalence can preserve certain types of properties, but these have to be categorical in nature (must be defined in terms of the module category and not the base ring).

Projectivity and injectivity are obvious examples that get preserved (because these types of modules can be characterized by a universal property). It turns out that finite-generability of a module can also be categorically characterized. In particular, $M$ is finitely-generated iff for any family $\{M_i\}_{i \in I}$ of modules such that $M = \sum_{i \in I} M_i$, there is some finite $J \subseteq I$ so that $M = \sum_{j \in J} M_j$.

In general, right regular modules are not preserved.

The number of elements needed to generate a f.g. module is also not a categorical quantity. However, the functors used to construct the Morita equivalence sends f.g. $S$-modules to cyclic $R$-modules. By the characterization of ideals in the matrix ring, this implies the following theorem.

Theorem 17.24

[theorem] Let $S$ be a PIRD. Then for any $n \ge 1$, $R = M_n(S)$ is a PIRD. [/theorem]

Definition (morita equivalence)

[definition] A ring $T$ is morita equivalent to $S$ if there is a categorical equivalence $\operatorname{mod}{-}T \cong \operatorname{mod}{-}S$. [/definition]

Definition (progenerator)

[definition] We say that $P_S$ is a progenerator if $P$ is f.g. projective and there is an epimorphism $n\cdot P \to S_S$ for some $n \ge 1$. [/definition]

An immediate example is the free modules $n\cdot S$. Indeed, the morphism $ns \mapsto s$ does the job. The reason this is important is because of the following result.

Theorem 17.25

[theorem] A ring $T$ is Morita equivalent to $S$ iff $T \cong \End_S(P)$, where $P_S$ is a progenerator in the category of $S$-modules. [/theorem]