Two results on modules whose endomorphism ring is semilocal. (Q1770890)

Let \(M_R\) be a right module over the ring \(R\). The authors establish two main results. The first of which, Theorem 2.3, is that if the dual Goldie dimension \(\text{codim(End}(M_R))\) of \(M_R\) is \(n\) and if \(M_R\) is isomorphic to a direct summand of a direct sum of finitely many modules \(A_i\), then \(M_R\) is isomorphic to a direct summand of a direct sum of \(m\) of the \(A_i\)'s where \(m\leq n\). The second key result, Theorem 3.4, is the following version of the Krull-Schmidt Theorem. If \(M_R=\bigoplus_{i\in I}M_i=\bigoplus N_{j\in J}\) are two direct sum decompositions of \(M_R\) into indecomposable \(\aleph_0\)-small quasi-small direct summands and if all the endomorphism rings \(\text{End}(M_k)\) and \(\text{End}(N_\ell)\) are homogeneous semilocal, than there is a one-to-one correspondence \(\varphi\colon I\to J\) such that \(M_i\cong N_{\varphi(i)}\) for each \(i\in I\). -- That this result is true when \(I\) and \(J\) are finite was shown by Barioli, Facchini, Raggi, and Ríos in 2001 and is reproved in this work using theory developed in Chapter 2. As corollaries, we can apply Theorem 3.4 when the \(M_i\)'s and \(N_j\)'s are finitely generated modules, or when the \(M_i\)'s and \(N_j\)'s are Artinian modules.