All rings considered in this paper are commutative with identity.
Recall from [D. F. Anderson and A. Badawi, Commun. Algebra 39, No. 5, 1646–1672 (2011; Zbl 1232.13001)], that if \( n\in \mathbb{N}\) and \(I\) is an ideal of a ring \(R\) then \(I\) is said to be an \(n\)-absorbing ideal of \(R\) if for any \(x_{1},\dots,x_{n+1}\in R\) such that \( x_{1}\dots x_{n+1}\in R,\) there are \(n\) of the \(x_{i}\)’s whose product is in \( I. \) The ideal \(I\) is said to be a strongly \(n\)-absorbing ideal of \(R\) if for any ideals \(I_{1},\dots,I_{n+1}\) of \(R\) such that \(I_{1}\dots I_{n+1}\subset I,\) there are \(n\) of the \(I_{i}\)’s whose product is contained in \(I.\) Two conjectures araised from these definitions:
Conjecture 1. Given \(n\in \mathbb{N},\) every \(n\)-absorbing ideal of \(R\) is also a strongly \(n\)-absorbing ideal of \(R.\)
Conjecture 2. Given \(n\in \mathbb{N},\) if \(I\) is an \(n\)-absorbing ideal of \(R\) then \(I[X]\) is an \(n\)-absorbing ideal of \(R[X].\)
In the paper under review the author proves that these two conjectures are true for any \(n\in \mathbb{N},\) when \(R\) is a locally divided ring or \(R\) is a \(2\)-AB ring. Moreover, he shows that for a ring \(R,\) there exists \(n\) such that every ideal of \(R\) is \(n\)-absorbing if and only if \(R\) is a strongly Laskerian ring with Krull dimension zero.
The author studies also finite absorbing rings. Recall that a ring \(R\) is finite absorbing if each ideal of \(R\) is an \(n\)-absorbing for a certain \( n\in \mathbb{N}.\) He proves that if \(R\) is a locally divided ring then \(R\) is finite absorbing if and only if \(R\) is strongly Laskerian.
Finally, finite absorbing rings on idealization of modules over rings are investigated. More precisely, and among other results, the author shows that for an integral domain \(R\) and a divisible \(R\)-module \(M,\) the ring \(R(+)M\) is finite absorbing if and only if \(M\) is a torsion free semisimple module and \(R\) is finite absorbing.