In this paper, given two quantales non necessary with identity. We investigate the ideals, prime ideals, radical ideals, primary ideals, and maximal ideals of the direct product. Unlike the case where those quantale are unital, an ideal (or primary ideal, or maximal ideal) of the their direct product need not be a sub-product (Lemma 3.1) of ide- als. The Theorem 4.2 extends the result on [3] for the product of two quantales.
