Let D be an integral domain with quotient field K,$mathcal{I}(D)$ be the set of nonzero ideals of D, and $w$ be the star-operation on D defined by $I_w={x{in}K{mid}xJ{subseteq}I$ for some $J{in}mathcal{I}(D)$ such that J is finitely generated and $J^{-1}=D}$. The D is called a Pr$ddot{u}$fer $v$-multiplication domain if $(II^{-1})_w=D$ for all nonzero finitely generated ideals I of D. In this paper, we show that D is a Pr$ddot{u}$fer $v$-multiplication domain if and only if $(A{cap}(B+C))_w=((A{cap}B)+(A{cap}C))_w$ for all $A,B,C{in}mathcal{I}(D)$, if and only if $(A(B{cap}C))_w=(AB{cap}AC)_w$ for all $A,B,C{in}mathcal{I}(D)$, if and only if $((A+B)(A{cap}B))_w=(AB)_w$ for all $A,B{in}mathcal{I}(D)$, if and only if $((A+B):C)_w=((A:C)+(B:C))_w$ for all $A,B,C{in}mathcal{I}(D)$ with C finitely generated, if and only if $((a:b)+(b:a))_w=D$ for all nonzero $a,b{in}D$, if and only if $(A:(B{cap}C))_w=((A:B)+(A:C))_w$ for all $A,B,C{in}mathcal{I}(D)$ with B, C finitely generated.