Abstract
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, includ-
ing 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 fur-
ther exploration, serving as a natural continuation of this article.