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Definition (categorical property)

[definition] A property $P$ on objects (resp. morphisms) in a module category $\operatorname{Mod}{-}R$ is said to be a categorical property if, for any category equivalence $F:\operatorname{Mod}{-}R \to \operatorname{Mod}{-}S$, whenever $M \in \operatorname{Mod}{-}R$ (resp. $g \in \Hom_{R}(M,N)$) satifies $P$, then so does $F(M)$ (resp. $F(g)$). [/definition]

Examples of this include morphisms being monomorphisms and epimorphisms. Because module categories are abelian categories, we know that $g$ being a monomorphism is characterized by having a trivial kernel. Similarly, $g$ being an epimorphism is characterized by having a trivial cokernel.

On the level of modules $M \in \operatorname{Mod}{-}R$, certainly the following are categorical properties:

• $M = 0$ and $M \neq 0$
• $M$ is simple
• $M$ is semisimple
• $M$ is indecomposable
• $M$ is uniform
• $M$ is noetherian
• $M$ is artinian
• $M$ has uniform dimension $n$
• $M$ has composition length $n$
• $M$ is f.g.

A submodule $N \subseteq M$ being maximal, minimal, essential, superfluous in $M$, or being a direct summand or a complement in $M$ are categorical.

Definition (morita equivalence)

[definition] Two rings $R, S$ are said to be Morita equivalent ($R \approx S$ for short) if there exists a category equivalence $F:\operatorname{Mod}{-}R \to \operatorname{Mod}{-}S$. A ring-theoretic property $P$ is said to be a Morita invariant if, whenever $R$ has the property $P$, so does every $S \approx R$. [/definition]

It turns out that if $S \approx R$, then $R{-}\operatorname{Mod} \cong S{-}\operatorname{Mod}$ as well. As such, there is no need to say right/left Morita equivalence.

Some examples of Morita invariants are semisimplicitiy, right noetherian/artinian, right (semi)hereditary, von Neumann regular.

Some non-examples are commutativity, local, reduced, domain, division ring, and even IBN (due to Bergman).

Recall that $R$ is Morita equivalent to its square matrix rings, we can lift the corresponding Morita invariants to the matrix ring and vice versa.