On $n$-absorbing prime ideals of commutative rings

Authors : Mohammed ISSOUAL, Najib MAHDOU, Moutu ABDOU SALAM MOUTUİ

Pages : 455-465

Doi:10.15672/hujms.816436

View : 17 | Download : 4

Publication Date : 2022-04-01

Article Type : Research Paper

Abstract :This paper investigates the class of rings in which every n n -absorbing ideal is a prime ideal, called n n -AB ring, where n n is a positive integer. We give a characterization of an n n -AB ring. Next, for a ring R R , we study the concept of Ω insert ignore into journalissuearticles values( R ); = { ω R insert ignore into journalissuearticles values( I ); ; I   is a proper ideal of  R } , Ωinsert ignore into journalissuearticles values(R);={ωRinsert ignore into journalissuearticles values(I);;I is a proper ideal of R}, where ω R insert ignore into journalissuearticles values( I ); = min { n ; I   is an  n -absorbing  ideal of  R } ωRinsert ignore into journalissuearticles values(I);=min{n;I is an n-absorbing ideal of R} . We show that if R R is an Artinian ring or a Prüfer domain, then Ω insert ignore into journalissuearticles values( R ); ∩ N Ωinsert ignore into journalissuearticles values(R);∩N does not have any gaps insert ignore into journalissuearticles values(i.e., whenever n ∈ Ω insert ignore into journalissuearticles values( R ); n∈Ωinsert ignore into journalissuearticles values(R); is a positive integer, then every positive integer below n n is also in Ω insert ignore into journalissuearticles values( R ); Ωinsert ignore into journalissuearticles values(R); );. Furthermore, we investigate rings which satisfy property insert ignore into journalissuearticles values(**); insert ignore into journalissuearticles values(i.e., rings R R such that for each proper ideal I I of R R with ω R insert ignore into journalissuearticles values( I ); < ∞ ωRinsert ignore into journalissuearticles values(I);<∞ , $\omega_{R}insert ignore into journalissuearticles values(I);=\mid Min_Rinsert ignore into journalissuearticles values(I);\mid $ ωRinsert ignore into journalissuearticles values(I);=∣MinRinsert ignore into journalissuearticles values(I);∣ , where M i n R insert ignore into journalissuearticles values( I ); MinRinsert ignore into journalissuearticles values(I); denotes the set of prime ideals of R R minimal over I I );. We present several properties of rings that satisfy condition insert ignore into journalissuearticles values(**);. We prove that some open conjectures which concern n n -absorbing ideals are partially true for rings which satisfy condition insert ignore into journalissuearticles values(**);. We apply the obtained results to trivial ring extensions.
Keywords : n absorbing ideal, prime ideal, primary ideal, Noetherian ring, Artinian ring, Prüfer ring