On n-absorbing prime ideals of commutative rings

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