AB5*-modules with the exchange property

Abstract

We prove that every $AB{5}^{\ast }$-module has an indecomposable decomposition. As an immediate consequence, every $AB{5}^{\ast }$-module M with the finite exchange is clean and has the full exchange. Moreover, in this case the module M admits an indecomposable decomposition $M={\oplus }_{i\in I}{M}_i$ with each M_i having local endomorphism ring, and the decomposition complements direct summands.

Keywords:

• AB$5^{\ast }$-modules
• clean
• discrete rings and modules
• exchange

2010 Mathematics Subject Classification:

• Primary: 16D40
• 16D50
• 16D60
• Secondary: 16L30
• 16L60
• 16P20
• 16P40
• 16P60

Acknowledgements

The second author would like to acknowledge the support received from the Ohio State University at Lima in the form of a research leave during the spring semester of 2019.

Additional information

Funding

The second author would like to acknowledge the support received from the Ohio State University at Lima in the form of a research leave during the spring semester of 2019.