Some properties of the weakly S-prime elements and the minimal weakly S-prime elements are studied. An equivalent description of the weakly S-prime elements is given. It is proved that a weakly S-prime element is preserved by the map between the quantales satisfying certain conditions and a two-sided element is weakly S-prime if and only if it is prime when S has the left absorbency. It is obtained that a maximal element is weakly S-prime in an idempotent two-sided quantale. At the same time, the concept of essential weakly S-prime element is introduced.The relations between weakly S-prime element and essential weakly S-prime elements are then discussed.

quantale;cyclic element;weakly S-prime element;minimal weakly S-prime element;essential weakly S-prime element