TITLE: Do the isomorphism classes of indecomposable objects in $R{\text{-mod}}$ form a set?
QUESTION [10 upvotes]: 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.

REPLY [12 votes]: 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.