Roth's theorems and decomposition of modules

AbstractIt is shown that there is a connection between Roth's theorems on similarity and equivalence of block-triangular matrices and decomposition of modules. The module property is that if M≅N⊕MN, then N is a summand of M. This holds for any commutative ring if M is finitely presented. New proofs of Roth's theorems are given for commutative rings. Some results are established in the noncommutative case