Krull-Schmidt theorem and semilocal endomorphism rings (Q2738454)

Throughout, ring means an associative ring with identity \(1\neq 0\). If \(R\) is a ring, by \(J(R)\) is denoted the Jacobson radical of \(R\). All modules are unital right modules. A ring \(R\) is said to be semilocal if \(R/J(R)\) is semisimple Artinian. A ring \(R\) is said to be homogeneous semilocal if \(R/J(R)\) is simple Artinian.NEWLINENEWLINENEWLINEThe author makes a survey of some very recent results on the Krull-Schmidt theorem. The properties of some classes of modules whose endomorphism ring is semilocal and their direct sum decompositions are described. In particular, he presents some results proved by Corisello, Barioli, Herbera, Raggi, Rios and Facchini about homogeneous semilocal rings and the Krull-Schmidt theorem for modules whose endomorphism ring is homogeneous semilocal.NEWLINENEWLINEFor the entire collection see [Zbl 0963.00025].