In this paper, given a quantale Q non necessary with identity. We establish that there exists a bijective correspondence between the set of all ideal of a quantale Qand the set of its all two-sided elements (see 4.3). We investigate the ideals, prime ideals and maximal ideals of Q1 × Q2. After having studied in all the contours the prime ideals, we introduce the spectrum topology, well known in rings theory as being Zariski topology.
AMS subject classification: 06F07, 18D10