Core and strongly core orthogonal elements in rings with involution

Abstract

In this paper, we present a new concept of the core orthogonality for two core invertible elements a and b in the rings with involution. a is said to be core orthogonal to b if a#b=0 and ba#=0, where a# is the core inverse of a. We study its characterizations and the core additivity. And the connection between the core orthogonality and core partial order have been given. We also give their matrix representations. Applying the core orthogonality, the concept of the strongly core orthogonality is defined and characterized. Moreover, we show that two arbitrary complementary projections are strongly core orthogonal.