On higher order z-ideals and z∘-ideals in commutative rings

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran.

Abstract

A ring R is called radically z-covered (resp. radically z∘-covered) if every z-ideal (resp. z∘-ideal) in R is a higher order z-ideal (resp. z∘-ideal). In this article we show with a counter-example that a ring may not be radically z-covered (resp. radically z∘-covered). Also a ring R is called z∘-terminating if there is a positive integer n such that for every m≥n, each z∘m-ideal is a z∘n-ideal. We show with a counter-example that a ring may not be z∘-terminating. It is well known that whenever a ring homomorphism ϕ:R→S is strong (meaning that it is surjective and for every minimal prime ideal P of R, there is a minimal prime ideal Q of S such that ϕ−1[Q]=P), and if R is a z∘-terminating ring or radically z∘-covered ring then so is S. We prove that a surjective ring homomorphism ϕ:R→S is strong if and only if ker(ϕ)⊆rad(R).

Keywords

References

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Algebraic Structures and Their Applications
Volume 11, Issue 1
February 2024
Pages 55-61

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