In this paper we shall formulate necessary and sufficient conditions for a semiprime ring to be both left and right Goldie in terms of the symmetric notion of biideal instead of one-sided ideals. We use this result to characterize semiprime Goldie rings with involution by conditions consistent with the notion of involution rings, that is, in terms of ascending chain condition for annihilator *-biideals and of maximum condition for *-biideal direct sums.