In this paper we study the Martindale ring of α-quotients Q associated with the partial action (R,α). Among other results we extend the partial action to Q and prove that it can be identified with an ideal of Q, the Martindale ring of β-quotients of T , where (T, β) denotes the enveloping action of (R,α). We prove that, in general, (Q, β) is not the enveloping action of (Q,α) and study the relationship between the rings R, Q, T and Q. Finally, we establish some properties related to the center of Q and the extended α-centroid of R.