Core–EP orthogonal elements in a ring with involution

Our goal is to study a new concept of the core–EP orthogonality for elements of a ring with involution, extending this concept presented for Drazin invertible bounded linear operators and the concept of the core orthogonality for elements of a ring with involution. Various properties and characterizations of core–EP orthogonal elements are proved as well as their matrix representation forms. For a pair of core–EP orthogonal elements, we propose equivalent conditions for the core–EP additivity to hold. We apply the core–EP orthogonality to introduce and study the strong core–EP orthogonality for elements of a ring with involution. Thus, some results for operators are generalized and a number of new results are given.