  When $R$ is a Noetherian ring and we have a family of ideals in which every
ideal contains at least one nonzero divisor, then it is already known that the
defining ideal of the multi-Rees algebra of these ideals is equal to a
saturated ideal. In such a case to get the defining ideal of the multi-Rees
algebra we only need to saturate the first syzygies of direct sum of this
family of ideals. However, this fact is not true, when at least one of these
ideals does not contain any nonzero divisor. In this paper we show that the
defining ideal of the multi-Rees algebra of a family of ideals of a polynomial
ring over a principal ideal ring, is equal to another kind of saturated ideal,
where we saturate more explicit polynomials other than just first syzygies.
Please notice that in general some of these ideals may not contain nonzero
divisors. Given this explicit formula, we can compute Gr\"{o}bner basis of the
defining ideal using an elimination order, which we talk about in the present
paper.
