Structure of zero-divisors to go up to related ideals

There are many ways for zero-dividing polynomials to go up to zero-dividing ideals when the base rings are IFP. The importance of these in ring theory leads us to consider the following ring conditions and study new useful roles of matrices for ring theory. Let R be a ring and a,b is an element of R\{0}. The first is the condition (*) that if ab = 0 then Ib = 0 for some nonzero ideal I subset of RaR of R or aJ = 0 for some nonzero ideal J subset of RbR of R. It is shown that from given any IFP ring, there can be constructed a non-IFP ring with the condition (*). We prove that a semiprime ring R with the condition (*) is both right and left nonsingular. The second is the condition (**) that if ab = 0 then IJ = 0 for some nonzero ideals I subset of RaR and J subset of RbR of R. We prove that every ring can be a subring of rings with the condition (**), that if R is an irreducible ring with the condition (**) then R is either a domain or non-semiprime, and that the condition (**) passes to polynomial rings when the base ring is semiprime. Various sorts of examples are given to illustrate and delimit the results obtained.