Multi-Rees algebras on principal ideal rings

Abstract: 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.