Condensed rings with zero-divisors

A commutative ring R with identity is condensed ( respectively strongly condensed) if for each pair of ideals I, J of R, IJ = {ij vertical bar i is an element of I, j is an element of J} (resp., IJ = iJ for some i is an element of I or IJ = Ij for some j is an element of J). In a similar fashion we can de. ne regularly condensed and regularly strongly condensed rings by restricting I and J to be regular ideals. We show that an arbitrary product of rings is condensed if and only if each factor is so, and that R[X] is condensed if and only if R is von Neumann regular. A number of results known in the domain case are extended to the ring case. Regularly strongly condensed and one-dimensional regularly condensed Noetherian rings are characterized.