Is there a set of isomorphism classes of indecomposable $R$-modules?

Question

Let $R$ be an associative ring with one.

Is there a cardinal $\kappa$ such that each indecomposable $R$-module has cardinality less than $\kappa\ ?$

Equivalently, is there a set $S$ of indecomposable $R$-modules such that any indecomposable $R$-module is isomorphic to some element of $S\ ?$

Answer 1

Shelah proved that there are indecomposable abelian groups of arbitrary cardinality. This had previously been proved by Fuchs for cardinalities smaller than the first inaccessible cardinal and (later) smaller than the first measurable cardinal.

Shelah, Saharon, Infinite Abelian groups, Whitehead problem and some constructions, Isr. J. Math. 18, 243-256 (1974). ZBL0318.02053.