Leti1,…,igH1…,Hf(g≥1,f≥1) be ideal in a a Noetherian ring R,letj1…,jgbe poitiveintegers,and let be an element in Ij1i(i=1,…,g). then b1…,bgare a superficial set of element of degree j1…,jgforI1…,Ig; H11…,hfin case there exist positve interers c1…,cgsuch thatforH=1…,for allki≥ciand for all ni≥0(i=1,…,g) In this paper we show the existence of such sets of elements, characterize them in sevaral ways(when eitherf=1 and H1=R Or Rad(I1…I1)⊆Rad(H1…,Hh)), show a few of their basic properties and use them to show that if each Iiis regular then for all ideals H in R,for all large kiand for allni≥(i=1,…,g)