Due to Bell, a ring R is usually said to be IFP if ab = 0 implies aRb = 0 for $a,b{in}R$. It is shown that if f(x)g(x) = 0 for $f(x)=a_0+a_1x$ and $g(x)=b_0+{cdots}+b_nx^n$ in R[x], then $(f(x)R[x])^{2n+2}g(x)=0$. Motivated by this results, we study the structure of the IFP when proper ideals are taken in place of R, introducing the concept of insertion-of-ideal-factors-property (simply, IIFP) as a generalization of the IFP. A ring R will be called an IIFP ring if ab = 0 (for $a,b{in}R$) implies aIb = 0 for some proper nonzero ideal I of R, where R is assumed to be non-simple. We in this note study the basic structure of IIFP rings.