Block Decomposition For Modules

Abstract

Block decomposition for rings has been introduced andshown to be unique in the literature (see [T. Y. Lam, GraduateTexts in Mathematics, 131, Springer-Verlag, New York, 1991]).Applying annihilator submodules, we extend this definition tomodules and show that every module $M$ has a unique blockdecomposition $M=\bigoplus_{i=1}^nM_i$ where each $M_i$ is anannihilator submodule. We also show that the block decompositionfor any ring $R$ and the block decomposition for the module $R_R$, are identical. Block decomposition provides us with a decomposition for $\edmp{M}$ because $\edmp{M}\iso\prod_{i=1}^n\edmp{M_i}$.