SUFFICIENT CONDITION FOR A NIL-CLEAN ELEMENT TO BE CLEAN IN A CERTAIN SUBRING OFM3(Z)

Abstract

Diesl has proved that a nil-clean ring is clean. However, not all nil-clean element of any ring is clean as showed by Andrica by providing counter examples in 2 x 2 matrices over Z. The objective of this study is to determine sufficient condition for a nil-clean element tobe clean in a certain subring of M3(Z). The two main methods are constructing certain subring, namely X3Z , of M3(Z) and then identifying idempotent and nilpotent elements in X3(Z) . This construction provides examples as the extension of those matrices founded by Andrica in the sense of matrix order and the different form of those matrices. The methods are used in finding the sufficient condition for nil-clean elements to be clean in a certain subring of M3(Z). By this finding, we follow up the previous researches especially from Diesl and Andrica. As the application, it is provided nil-clean elements in X3(Z) which are clean and some other elements which are not clean.