DIRECT SUMS OF INFINITELY MANY KERNELS

Abstract

AbstractLet 𝒦 be the class of all rightR-modules that are kernels of nonzero homomorphisms φ:E1→E2for some pair of indecomposable injective rightR-modulesE1,E2. In a previous paper, we completely characterized when two direct sumsA1⊕⋯⊕AnandB1⊕⋯⊕Bmof finitely many modulesAiand Bjin 𝒦 are isomorphic. Here we consider the case in which there are arbitrarily, possibly infinitely, manyAiandBjin 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 rightR-modules (a module is uniserial when its lattice of submodules is linearly ordered).