Finitary ideals of direct products in quantales

Abstract

The notion of quantale, which designates a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins, appears in various areas
of mathematics-in quantaloid theory, in non classical logic as completion of the Lindebaum algebra, and in different representations of the spectrum of a C∗ algebra as
many-valued and non commutative topologies. To put it briefly, its importance is no
longer to be demonstrated. Quantales are ring-like structures in that they share with
rings the common fact that while as rings are semi groups in the tensor category of
abelian groups, so quantales are semi groups in the tensor category of sup-lattices.
In 2008 Anderson and Kintzinger [1] investigated the ideals, prime ideals, radical ideals, primary ideals, and maximal of a product ring R × S of two commutative
non non necceray unital rings R and S: Something resembling rings are quantales by
analogy with what is studied in ring, we begin an investigation on ideals of a product
of two quantales. In this paper, given two quantales Q1 and Q2; not necessarily with
identity, we investigate the ideals, prime ideals, primary ideals, and maximal ideals of
the quantale Q1 × Q2