Modules whose Endomorphism Rings have Finite Triangulating Dimension and Some Applications

Abstract

By defining orthogonal decomposition for modules, we prove that an R-module M has only finitely many fully invariant direct summands if and only if End R (M) has triangulating dimension \({n = {\rm Sup}\{k \in \mathbb{N} | M = \oplus^{k}_{i=1}M_{i}}\) is left orthogonal}. Denoting n = τdim(M R ), the triangulating dimension of M R , it is shown that τ dim(M R ) is Morita invariant, and when R is an Artinian principal ideal ring, τ dim(M R ) is the number of socle components of M R . If R is commutative then R is perfect (resp. a finite direct product of domains) if and only if it is semi-Artinian (resp. semiprime extending) with finite triangulating dimension. A recent result of Birkenmeier et al. [4] is generalized into a module setting.