S-VERSIONS AND S-GENERALIZATIONS OF IDEMPOTENTS, PURE IDEALS AND STONE TYPE THEOREMS

Abstract

Let R be a commutative ring with nonzero identity and M be an R -module. In this paper, we first introduce the concept of Sidempotent element of R. Then we give a relation between S-idempotents of R and clopen sets of S-Zariski topology. After that we define S -pure ideal which is a generalization of the notion of pure ideal. In fact, every pure ideal is S -pure but the converse may not be true. Afterwards, we show that there is a relation between S -pure ideals of R and closed sets of S-Zariski topology that are stable under generalization.