On 2r-ideals in commutative rings with zero-divisors

In this article, we are interested in uniformly prpr-ideals with order ≤2\le 2 (which we call 2r2r-ideals) introduced by Rabia Üregen in [On uniformly pr-ideals in commutative rings, Turkish J. Math. 43 (2019), no. 4, 18781886]. Several characterizations and properties of these ideals are given. Moreover, the comparison between the (nonzero) 2r2r-ideals and certain classes of classical ideals gives rise to characterizations of certain rings based only on the properties of the ideals consisting only of zero-divisors. Namely, among other things, we compare the class of (nonzero) 2r2r-ideals with the class of (minimal) prime ideals, the class of minimal prime ideals and their squares, and the class of primary ideals. The study of 2r2r-ideal in polynomial rings allows us to give a new characterization of the rings satisfying the famous AA-property.