Let R be a ring, and ${alpha}$ be an endomorphism of R. An additive mapping H : R ${ ightarrow}$ R is called a left ${alpha}$-multiplier (centralizer) if H(xy) = H(x)${alpha}$(y) holds for all x,y $in$ R. In this paper, we shall investigate the commutativity of prime and semiprime rings admitting left ${alpha}$-multiplier satisfying any one of the properties: (i) H([x,y])-[x,y] = 0, (ii) H([x,y])+[x,y] = 0, (iii) $H(x{circ}y)-x{circ}y=0$, (iv) $H(x{circ}y)+x{circ}y=0$, (v) H(xy) = xy, (vi) H(xy) = yx, (vii) $H(x^2)=x^2$, (viii) $H(x^2)=-x^2$ for all x, y in some appropriate subset of R.