Block Decomposition For Modules

Block decomposition for rings has been introduced and shown to be unique in the literature (see [T. Y. Lam, Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 1991]). Applying annihilator submodules, we extend this definition to modules and show that every module M has a unique block decomposition M = Ln i=1 Mi where each Mi is an annihilator submodule. We also show that the block decomposition for any ring R and the block decomposition for the module RR, are identical. Block decomposition provides us with a decomposition for End(M) because End(M) ∼= Qn i=1 End(Mi).