Abstract. Due to Bell, a ring R is usually said to be IFP if ab = 0implies aRb = 0 for a;b 2R. It is shown that if f(x)g(x) = 0 for f(x) =a 0 +a 1 x and g(x) = b 0 + +b n x 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 properideals are taken in place of R, introducing the concept of insertion-of-ideal-factors-property (simply, IIFP) as a generalization of the IFP. Aring R will be called an IIFP ring if ab = 0 (for a;b 2R) implies aIb = 0for 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. 1. IntroductionInsertion-of-Factors-Property has done important roles in noncommutativering theory and module theory. Throughout this note every ring is an asso-ciative ring with identity unless otherwise stated. Given a ring R, let N(R)and N (R) denote the set of all nilpotent elements and the prime radical in R,respectively. The polynomial ring with an indeterminate xover Ris denotedby R[x]. The nby nfull (resp. upper triangular) matrix ring over Ris denotedby Mat