Abstract
Let R be a unital ring with involution. In this paper, we first show that for an element a 2 R, a is Moore-Penrose invertible if and only if a is well-supported if and only if a is co-supported. Moreover, several new necessary and sufficient conditions for the existence of the Moore-Penrose inverse of an element in a ring R are obtained. In addition, the formulae of the Moore-Penrose inverse of an element in a ring are presented.
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