Block Decomposition For Modules

Year 2017, Volume: 22 Issue: 22, 187 - 201, 11.07.2017

H. Khabazian

https://doi.org/10.24330/ieja.325944

Cited By: 1

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

• J. A. Beachy and W. D. Blair, Rings whose faithful left ideals are cofaithful, Pacific J. Math., 58(1) (1975), 1-13.
• K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommuta- tive Noetherian Rings, Second Edition, London Mathematical Society Student Texts, 61, Cambridge University Press, Cambridge, 2004.
• H. Khabazian, Existence and uniqueness of a certain type of subdirect product, to appear in Casp. J. Math. Sci.
• T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 1991.