Taking a ring-theoretic perspective as our motivation, the main aim of this series is to establish a comprehensive theory of ideals in commutative quantales with an identity element. This particular article focuses on an examination of several key properties related to ideals in quantale, including prime, semiprime, radical, primary, irreducible, and strongly irreducible ideals. Furthermore, we investigate the primary decomposition problem for quantale ideals. In conclusion, we present a set of future directions for further exploration, serving as a natural continuation of this article.
We study the topology of a class of proper submodules and some of its distinguished subclasses and call them structure spaces. We give several criteria for the quasi-compactness of these structure spaces. We study T_0 and T_1 separation properties and characterize structure spaces in which nonempty irreducible closed subsets have unique generic points. We provide a sufficient condition for the connectedness of structure spaces. We prove that the structure spaces of proper submodules are spectral, and moreover, we characterize the spectral structure spaces of Noetherian modules. Finally, we discuss continuous functions between these spaces.
