On S-1-absorbing prime and weakly S-1-absorbing prime ideals
Let A be a commutative ring with identity and S be a multiplicative subset of A. In this paper, we introduce and study the notions of S-1-absorbing and weakly S-1-absorbing prime ideals as generalizations of the notion of prime ideals. We define a proper ideal I disjoint with S to be an S-1-absorbing (resp. a weakly S-1-absorbing) prime ideal if there exists s ∈ S such that for all nonunit elements a, b, c ∈ A such that abc ∈ I (resp. 0 ≠ abc ∈ I), we have sab ∈ I or sc ∈ I. Several properties and characterizations of S-1-absorbing prime and weakly S-1-absorbing prime ideals are given. Moreover, we study the transfer of the above properties to some constructions of rings such as trivial ring extensions and amalgamation of rings along an ideal.