S-primal ideals over commutative rings

Let [Formula: see text] be a commutative ring with unity [Formula: see text] and let [Formula: see text] be a multiplicative subset of [Formula: see text] with [Formula: see text]. Let [Formula: see text]. In this paper we introduce the concept of [Formula: see text]-primal ideal which is different from primal ideal. Let [Formula: see text] be a proper ideal of [Formula: see text] disjoint with [Formula: see text] (i.e. [Formula: see text]). An element [Formula: see text] is defined to be [Formula: see text]-prime to [Formula: see text], if for any [Formula: see text] with [Formula: see text], then there exists [Formula: see text] such that [Formula: see text]. An element [Formula: see text] is not [Formula: see text]-prime to [Formula: see text] if there exists [Formula: see text] with [Formula: see text], such that [Formula: see text] for all [Formula: see text]. We denote by [Formula: see text] the set of all elements in [Formula: see text] that are not [Formula: see text]-prime to [Formula: see text]. We define a proper ideal [Formula: see text] of [Formula: see text] to be [Formula: see text]-primal, if the set [Formula: see text] forms an ideal of [Formula: see text]. Many results and examples concerning [Formula: see text]-primal ideals are given.