Characterizations of EP, normal and Hermitian elements in rings using generalized inverses

The properties and some equivalent characterizations of equal projection (EP), normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP under the existence of core inverses are proposed. Let a∈R#, then a is EP if and only if aa#a#=a#aa#. At the same time, the equivalent characterizations of a regular element to be EP are discussed. Let a∈R, then there exist b∈R such that a=aba and a is EP if and only if a∈R#, a#=a#ba. Similarly, some equivalent conditions that an element is normal under the existence of core inverses are proposed. Let a∈R#, then a is normal if and only if a*a#=a#a*. Also, some equivalent conditions of normal and Hermitian elements in rings with involution involving powers of their group and Moore-Penrose inverses are presented. Let a∈R#∩ R#, n∈N, then a is normal if and only if a*a* (a#)n=a#a*(a*)n. The results generalize the conclusions of Mosi et al. © 2017, Editorial Department of Journal of Southeast University. All right reserved.