Abstract
Let $R$ be a unital ring with involution. The $(j,m)$-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 $(j,m)$-core invertible are given. Moreover, several additive and product properties of two $(j,m)$-core invertible elements are investigated and a order related to the $(j,m)$-core inverse is introduced.***********************************************************************************************************************************
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