S-primary ideals of a commutative ring

Abstract

Let R be a commutative ring with identity and $S⊈R$ a multiplicative subset. We define a proper ideal P of R disjoint from S to be S-primary if there exists an $s\in S$ such that for all $a,b\in R$ if $ab\in P$ then $sa\in P$ or $sb\in \sqrt{P}.$ We show that S-primary ideals enjoy analogs of many properties of primary ideals and we study the form of S-primary ideals of the amalgamation of A with B along an ideal J with respect to f (denoted by $A\mathrm{}{\bowtie }^f\mathrm{}J$), introduced and studied by D’Anna et al. S-primary ideals of the form $I(+)M$ of the trivial ring extensions and S-primary ideals of the form $I{\bowtie }^{f,g}(J,J\prime )$ and ${(K,L)}^{f,g}$ of the bi-amalgamations $A{\bowtie }^{f,g}(J,J\prime )$ are characterized.

Keywords:

• Amalgamated algebra
• primary ideal
• S-primary ideal
• S-prime ideal

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A15
• 13B02
• 13B10
• 13B25
• 13B30
• 13F20

Acknowledgments

I would like to thank the anonymous referee for his/her thorough review and useful comments that helped improve the clarity and the relevance of this paper. Also I am very grateful to Professor Noômen Jarboui for his many helpful suggestions and comments.