S-primal ideals over commutative rings

Abstract

Let R be a commutative ring with unity (1=0) and let S be a multiplicative subset of R with 1 S. Let S0 = S\{0}. In this paper we introduce the concept of S-primal ideal which is different from primal ideal. Let I be a proper ideal of R disjoint with S0 (i.e. I S0 = θ). An element a R is defined to be S-prime to I, if for any r R with ra I, then there exists s S0 such that sr I. An element a R is not S-prime to I if there exists r R with ar I, such that ϵ I for all s S0. We denote by νS(I) the set of all elements in R that are not S-prime to I. We define a proper ideal I of R to be S-primal, if the set νS(I) forms an ideal of R. Many results and examples concerning S-primal ideals are given. © 2026 World Scientific Publishing Company.