科研之友
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摘要
Let a, d be two elements in rings and a(||d) be the inverse of a along d. When a(||d) exists, we obtain several characterizations for the invertibility of aa(||d) - a(||d)a, which is related to the invertibility of elements expressed by certain functions of a, d and suitable elements from the center of the ring. On the other hand, some equivalent conditions for the equality aa(||d) = a(||d)a, as the complement of the previous invertibility in some sense, are given by means of the group inverses and the ring units, respectively. Then, the results obtained are applied in a *-ring, namely, when d = a*, the co-EP and EP properties are deduced correspondingly.
