On (1,2)-absorbing primary ideals and uniformly primary ideals with order ≤ 2

This paper introduces a subset of the set of 1-absorbing primary ideals introduced in [3]. An ideal I of a ring R is (1,2)-absorbing primary if, whenever non-unit elements α, β, γ ∈ R with αβγ ∈ I,then αβ ∈ I or γ² ∈ I. The introduced notion is related to uniformly primary ideals introduced in [5]. The first main objective of this paper is to compare (1,2)-absorbing primary ideals with uniformly primary ideals with order less than or equal 2, as well as to characterize them in many classes of rings. The second part of this paper characterizes, by using (1,2)-absorbing primary ideals, the rings R for which all ideals lie between N(R) (the nil-radical of R)and N(R)₂.