Direct Sums of Infinitely Many Kernels

Abstract

Let K be the class of all right R-modules that are kernels of nonzero homomorphisms ϕ : 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 ⊕ · · · ⊕ A n and B 1 ⊕ · · · ⊕ 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).2000 Mathematics subject classification: primary 16D70; secondary 16E05, 16L30.

Direct products of modules whose endomorphism rings have at most two maximal ideals

“…Notice that in [5] it was proved that if { A i | i ∈ I } and { B j | j ∈ J } are two families of modules over a ring R, all the B j 's are kernels of non-injective morphisms between indecomposable injective modules and there exist bijections σ, τ :…”

Uniqueness of Decomposition, Factorisations, G-Groups and Polynomials

“…11. (Weak Krull-Schmidt Theorem for kernels of morphisms between indecomposable injective modules,[19, Theorem 2.7] and[14]) Let ϕ i : E i,0 → E i,1 (i = 1, 2, .. . , n) and ϕ j : E j,0 → E j,1 (j = 1, 2, .…”