Let $A$ be a ring and $\text{mod} A$ the category of finitely generated (right) modules over $A$. Is the class of isomorphism types of $\text{mod} A$ always a set? In particular, is it the case if $A$ is an Artin algebra?


EDIT: The answer is true for an arbitrary ring $A$. However, can we bound the cardinality if we know that $A$ is Artin algebra? Can the cardinality of an Artin algebra be bounded?



Is the class of isomorphism types of a module category always a set?
Yes, because you're only considering finitely generated modules. No assumptions on $A$ are required.


Indeed, a finitely generated module is always a quotient of some $A^n$ by some submodule. For each $n$, there is a set of submodules of $A^n$ (of size $\leq 2^{|A^n|}$), and there are countably many integers (by definition).


So there is a (very crude) bound of $\sum\_{n\in\mathbb N}2^{|A^n|}$ on the number of isomorphism classes of finitely generated $A$-modules; in particular there are "set-many" such isomorphism classes (more precisely : there is a set of finitely generated $A$-modules such that any finitely generated $A$-module is isomorphic to at least one in that set)


In fact, for any cardinal $\kappa$, there is a set of isomorphism classes of $\leq \kappa$-generated $A$-modules, the proof is exactly the same.


