ON THE ANDERSON-BADAWI ωR[X](I[X]) = ωR(I) CONJECTURE

06/30/2020

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by   Peyman Nasehpour, Ph.D., et al.

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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 x1 . . . xn+1 ∈ I for x1, . . . , xn+1 ∈ R, then there are n of the xi’s whose product is in I and conjecture that ωR[X] (I[X]) = ωR(I) for any ideal I of an arbitrary ring R, where ωR(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.