An embedding theorem on triangular matrix rings

Abstract

Every matrix in the triangular matrix ring over a bleached local ring R is similar to a ‘simple form’, which is, in most cases, contained in a subring of with isomorphic to a direct product of two triangular matrix rings over R of smaller size. This result, called the embedding theorem, suggests a new approach for handling triangular matrix rings. It is applied to proving several results on the strong cleanness and strong 2-sum property of triangular matrix rings.

Keywords:

• Triangular matrix ring
• bleached local ring
• strongly clean ring
• strong 2-sum property

AMS Subject Classifications:

• Primary: 16L30
• 16U60
• 16U99

Acknowledgements

We thank Professor T.Y. Lam for Definition 2.4. Part of the work was carried out when Zhou was visiting Guangxi Teachers Education University sponsored by the National Natural Science Foundation of China. He gratefully acknowledges the hospitality from the host university.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Tang was supported by the Natural Science Foundation of China [grant numbers 11161006, 11461010], and the Guangxi Science Research and Technology Development Project [grant number 1599005-2-13]. Zhou was supported by a Discovery Grant from NSERC of Canada.