On the Anderson-Badawi $\omega_{R[X]}(I[X])=\omega_R(I)$ conjecture

Abstract

On the Anderson-Badawi ω R[X] (I[X]) = ω R (I) conjectureArchivum Mathematicum, Vol. 52 (2016) ON THE ANDERSON-BADAWI ω R[X] (I[X]) = ω R (I) CONJECTURE Peyman NasehpourAbstract. Let R be a commutative ring with an identity different from zero and n be a positive integer. Anderson and Badawi, in their paper on n-absorbing ideals, define a proper ideal I of a commutative ring R to be an n-absorbing ideal of R, if whenever x 1 . . . x n+1 ∈ I for x 1 , . . . , x n+1 ∈ R, then there are n of the x i 's whose product is in I and conjecture that ω R [X] (I[X]) = ω R (I) for any ideal I of an arbitrary ring R, where ω R (I) = min{n : I is an n-absorbing ideal of R}. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions hold:(1) The ring R is a Prüfer domain.(2) The ring R is a Gaussian ring such that its additive group is torsion-free.(3) The additive group of the ring R is torsion-free and I is a radical ideal of R. IntroductionLet R be a commutative ring with an identity different from zero and n be a positive integer. Anderson and Badawi, in their paper [3], define a proper ideal I of a commutative ring R to be an n-absorbing ideal of R, if whenever x 1 . . . x n+1 ∈ I for x 1 , . . . , x n+1 ∈ R, then there are n of the x i 's whose product is in I. In the fourth section of their paper, they conjecture that ω R [X] (I[X]) = ω R (I) for any ideal I of an arbitrary ring R, where ω R (I) = min{n : I is an n-absorbing ideal of R}.Clearly a 1-absorbing ideal is just a prime ideal and it is a well-known result in commutative ring theory that I is a prime ideal of R iff I[X] is a prime ideal of R [X]. In [3, Theorem 4.15], it is also proved that I[X] is a 2-absorbing ideal of R[X] iff I is a 2-absorbing ideal of R.In this paper, we use content formula techniques to prove that their conjecture is true, i.e., ω R[X] (I[X]) = ω R (I) for an ideal I of R, if one of the following conditions hold:(1) The ring R is a Prüfer domain.2010 Mathematics Subject Classification: primary 13A15; secondary 13B02, 13B25, 13F05.

The Ohm–Rush content function II. Noetherian rings, valuation domains, and base change

“…(2) a semicontent algebra [ES16b] if it is faithfully flat and for any mul- [OR72] if it is faithfully flat and for any f, g ∈ S, there is some n ∈ N with c(f ) n c(g) = c(f ) n−1 c(f g), (4) a Gaussian algebra [Nas16a] if it is faithfully flat and for any f, g ∈ S,…”

“…The key idea is to generalize the notion of the "content" of a polynomial to an element of S and then see which formulas the function satisfies. In increasing order of specificity, one may ask (with some updated terminology) whether a faithfully flat Ralgebra S is (1) Ohm-Rush (from [OR72], terminology from [ES16b]), (2) weak content [Rus78], (3) semicontent [ES16b], (4) content [OR72], or (5) Gaussian [Nas16a].…”

Uniformly 2-absorbing primary ideals of commutative rings

“…Let I be a 2-absorbing primary ideal of R. Then P = √ I is a 2-absorbing ideal of R by [4,Theorem 2.2]. We say that I is a P -2-absorbing primary ideal of R. For more studies concerning 2-absorbing (submodules) ideals we refer to [5,9,10,15,16]. These concepts motivate us to introduce a generalization of uniformly primary ideals.…”

n-ABSORBING MONOMIAL IDEALS IN POLYNOMIAL RINGS

“…This conjecture originates from the well-known result that I is a prime ideal (i.e., 1-absorbing ideal) of R if and only if I[X] is a prime ideal of R[X]. Anderson and Badawi themselves proved this conjecture for an arbitrary commutative ring when n = 2 ([1, Theorem 4.15]), and Nasehpour proves that the second conjecture holds for every n ∈ N when R belongs to certain classes of rings, including the class of Prüfer domains ( [15]). In [11] Laradji independently proved that the second conjecture holds when R is an arithmetical ring, i.e., when the set of ideals of R M is totally ordered under set inclusion for each maximal ideal M of R.…”