Do the isomorphism classes of indecomposable objects in $R{\text{-mod}}$ form a set?

Question

Let $R$ be a unital (associative) ring. Consider the category $R\text{-mod}$ of unitary left $R$-modules. Set $\text{Indec}(R)$ to be the class of all isomorphism classes of indecomposable objects in $R\text{-mod}$.

Is $\text{Indec}(R)$ a set? If not, for which ring $R$ is $\text{Indec}(R)$ a set, and for which ring $R$ is $\text{Indec}(R)$ a proper class?

Clearly, if $R$ is semisimple, then $\text{Indec}(R)$ coincides with the set $\text{Irred}(R)$ of isomorphism classes of simple $R$-modules. I am not sure what would happen if $R$ is nonsemisimple.

If possible, I would like to request a reference that discusses a more general result. That is, if $\mathscr{C}$ is an arbitrary abelian category, and $\text{Indec}(\mathscr{C})$ is the class of all isomorphism classes of indecomposable objects in $\mathscr{C}$. How do we tell when $\text{Indec}(\mathscr{C})$ is a set?

Just as in the case of $R\text{-mod}$, the same observation holds: if $\mathscr{C}$ is semisimple, then $\text{Indec}(\mathscr{C})$ is identical to the class $\text{Irred}(\mathscr{C})$ of isomorphism classes of simple objects in $\mathscr{C}$. However, I am very certain that there are nonsemisimple abelian category $\mathscr{C}$ such that $\text{Indec}(\mathscr{C})$ is a proper class. Such an example is very welcome. $\phantom{aaa}$ Edit: With help from YCor, at least when $R=\mathbb{Z}$, $\text{Indec}(\mathbb{Z})$ is a proper class.

Remark. I am even more interested in the case where $R$ is a (not necessarily finite-dimensional) $\mathbb{K}$-algebra for some field $\mathbb{K}$. We may assume that $R$ is countable-dimesional. Even more specifically, I would like to know what happens if $R$ is the universal enveloping algebra of some (not necessarily finite-dimensional) Lie algebra $\mathfrak{g}$ over some field $\mathbb{K}$. Again, we may assume that $\mathfrak{g}$ is countable-dimensional. However, anything that can elaborate me on how to find out when $\text{Indec}(\mathscr{C})$ is a set will be greatly appreciated.

Answer 1

In Conjecture $1_{\infty}$ of

Simson, Daniel, On large indecomposable modules, endo-wild representation type and right pure semisimple rings., Algebra Discrete Math. 2003, No. 2, 93-118 (2003). ZBL1067.16029,

Simson conjectures that a right noetherian ring either

• is right pure semisimple (which would imply it is right artinian, and conjecturally would imply that it is right artinian of finite representation type), or
• has indecomposable modules of arbitrarily large cardinality.

So if this conjecture is true, then the answer to the question "for which noetherian rings is $\operatorname{Indec}(R)$ a proper class" is "almost all of them".

Simson proved this conjecture for several classes of finite dimensional algebras (e.g., finite dimensional local $k$-algebras with residue field $k$, and group algebras of finite groups) in

Simson, Daniel, On Corner type endo-wild algebras., J. Pure Appl. Algebra 202, No. 1-3, 118-132 (2005). ZBL1151.16014.

For non-noetherian rings the answer will be more complicated, since, for example, there are non-noetherian rings for which every indecomposable module is simple.