Invertibility in rings of the commutator ab – ba, where aba=a and bab=b

Abstract

Let ℛ be a ring and a, b∈ℛ satisfy aba = a and bab = b. We characterize when ab − ba is invertible. This study is specialized when ℛ has an involution and when b is the Moore–Penrose inverse of a.

Keywords:

• ring
• involution
• generalized inverses

Acknowledgements

We would like to thank the referee for his/her careful reading. The first author is supported by Spanish Project MTM2010-18539, the second author is supported by the National Natural Science Foundation of China (11601005) and the Ministry of Education Science and Technology Key Project (210164) and the third author by the Ministry of Science, Technology and Development, Republic of Serbia under Grant No. 174025.