Block Decomposition For Modules

Abstract

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 $M=\bigoplus_{i=1}^nM_i$ where each $M_i$ is an
annihilator submodule.  We also show that the block decomposition
for 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 $\edmp{M}\iso\prod_{i=1}^n\edmp{M_i}$.
 

Keywords

• Annihilator,block orthogonal

References

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4. T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 1991.