Finitary ideals of direct products in quantales

The notion of quantale, which designates a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins, appears in various areasof mathematics-in quantaloid theory, non classical logic as completion the Lindebaum algebra, and different representations spectrum C∗ algebra asmany-valued commutative topologies. To put it briefly, its importance is nolonger to be demonstrated. Quantales are ring-like structures that they share withrings common fact while rings semi groups tensor category ofabelian groups, so quantales sup-lattices.In 2008 Anderson Kintzinger [1] investigated ideals, prime radical primary maximal product ring R × S two commutativenon necceray unital S: Something resembling byanalogy what studied ring, we begin investigation on ideals productof quantales. In this paper, given Q1 Q2; not necessarily withidentity, investigate ofthe quantale Q2