Pseudo core inverses in rings with involution

Let $R$ be a ring with involution. In this paper, we introduce a new type of generalized inverse called pseudo core inverse in $R$. The notion of core inverse was introduced by Baksalary and Trenkler for matrices of index 1 in 2010 and then it was generalized to an arbitrary $*$-ring case by Raki\'{c}, Din\v{c}i\'{c} and Djordjevi\'{c} in 2014. Our definition of pesudo core inverse extends the notion of core inverse to elements of an arbitrary index in $R$. Meanwhile, it generalizes the notion of core-EP inverse, introduced by Manjunatha Prasad and Mohana for matrices in 2014, to the case of $*$-ring. Some equivalent characterizations for elements in $R$ to be pseudo core invertible are given and expressions are presented especially in terms of Drazin inverse and \{1,3\}-inverse. Then, we investigate the relationship between pseudo core inverse and other generalized inverses. Further, we establish several properties of the pseudo core inverse. Finally, the computations for pseudo core inverses of matrices are exhibited.