The $(j,m)$-core inverse in rings with involution

Let $R$ be a unital ring with involution. The $insert ignore into journalissuearticles values(j,m);$-core inverse of a complex matrix was extended to an element in $R$. New necessary and sufficient conditions such that an element in $R$ to be $insert ignore into journalissuearticles values(j,m);$-core invertible are given. Moreover, several additive and product properties of two $insert ignore into journalissuearticles values(j,m);$-core invertible elements are investigated and a order related to the $insert ignore into journalissuearticles values(j,m);$-core inverse is introduced. ***********************************************************************************************************************************