Direct sums of infinitely many kernels

Let K be the class of all right R-modules that are kernels of nonzero homomorphisms \varphi: E_1 -> E_2 for some pair of indecomposable injective right R-modules E_1, E_2. In a previous paper, we completely characterized when two direct sums A_1 \oplus\dots\oplus A_n and B_1 \oplus\dots\oplus B_m of finitely many modules A_i and B_j in K are isomorphic. Here we consider the case in which there are arbitrarily, possibly infinitely, many A_i and B_j in K. In both the finite and the infinite case, the behaviour is very similar to that which occurs if we substitute the class K with the class U of all uniserial right R-modules (a module is uniserial when its lattice of submodules is linearly ordered).