On regular near-rings with (m,n)-potent conditions

Jat and Choudhari defined a near-ring R with left bipotent or right bipotent condition in 1979. Also, we can define a near-ring R as subcommutative if aR=Ra for all a in R. From these above two concepts it is natural to investigate the near-ring R with the properties aR=Ra² (resp. a² R=Ra) for each a in R. We will say that such is a near-ring with (1,2)-potent condition (resp. a near-ring with (2,1)-potent condition). Thus, we can extend a general concept of a near-ring R with (m,n)-potent condition, that is, am R=Raⁿ for each a in R, where m, n are positive integers. We will derive properties of near-ring with (1,n) and (n,1)-potent conditions where n is a positive integer, any homomorphic image of (m,n)-potent near-ring is also (m,n)-potent, and we will obtain some characterization of regular near-rings with (m,n)-potent conditions.