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 nn-absorbing ideal is a prime ideal, called nn-AB ring, where nn is a positive integer. We give a characterization of an nn-AB ring. Next, for a ring RR, we study the concept of Ω(R)={ωR(I);I is a proper ideal of R},Ω(R)={ωR(I);I is a proper ideal of R}, where ωR(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 RR is an Artinian ring or a Prüfer domain, then Ω(R)∩NΩ(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 nn is also in Ω(R)Ω(R)). Furthermore, we investigate rings which satisfy property (**) (i.e., rings RR such that for each proper ideal II of RR with ωR(I)<∞ωR(I)<∞, $\omega_{R}(I)=\mid Min_R(I)\mid $ωR(I)=∣MinR(I)∣, where MinR(I)MinR(I) denotes the set of prime ideals of RR minimal over II). We present several properties of rings that satisfy condition (**). We prove that some open conjectures which concern nn-absorbing ideals are partially true for rings which satisfy condition (**). We apply the obtained results to trivial ring extensions.