AI Summary & Key Takeaways
AI InsightWe apply the obtained results to trivial ring extensions.
📌 3 Key Takeaways
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 ( R ) = { ω R ( I ) ; I is a proper ideal of R } , Ω(R)={ωR(I);I is a proper ideal of R}, where ω R ( I ) = min { n ; I is an n -absorbing ideal of R } ωR(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 ( R ) ∩ N Ω(R)∩N does not have any gaps (i.e., whenever n ∈ Ω ( R ) n∈Ω(R) is a positive integer, then every positive integer below n n is also in Ω ( R ) Ω(R) ). Furthermore, we investigate rings which satisfy property (**) (i.e., rings R R such that for each proper ideal I I of R R with ω R ( I ) < ∞ ωR(I)<∞ , $\omega_{R}(I)=\mid Min_R(I)\mid $ ωR(I)=∣MinR(I)∣ , where M i n R ( I ) MinR(I) denotes the set of prime ideals of R R minimal over I I ). We present several properties of rings that satisfy condition (**). We prove that some open conjectures which concern n n -absorbing ideals are partially true for rings which satisfy condition (**). We apply the obtained results to trivial ring extensions.