On the nil-clean matrix over a UFD

 In this paper we characterize all ۲\times ۲ idempotent and nilpotent matrices over an integral domain and then we characterize all ۲\times ۲ strongly nil-clean matrices over a PID. Also, we determine when a ۲\times ۲ matrix  over a UFD is nil-clean.

Keywords:

Rank of a matrix , Idempotent matrix , Nilpotent matrix , Nil-clean matrix , Strongly nil-clean matrix