Primary decomposition in a soft ring and a soft module

The main objective of this study is to swing Krull intersection theorem in primary decomposition of rings and modules to the primary decomposition of soft rings and soft modules. To fulfill this aim several notions like soft prime ideals, soft maximal ideals, soft primary ideals, and soft radical ideals are introduced for a soft ring over a given unitary commutative ring. Consequently, the primary decomposition of soft rings and soft modules is established. In addition, the ascending and descending chain conditions on soft ideals and soft sub modules of soft rings and soft modules are introduced, respectively, enabling us to develop the notions of soft Noetherian rings and soft Noetherian modules.