2026/07/20 by Ahmad Moussavi, Peter Danchev, Arash Javan +1
#math.RA #math.RT
We introduce and study the so-called \it generalized √(J)-fine rings, where every element outside the Jacobson radical is the sum of a unit and an element from the set √(J(R)) := \ x ∈ R : xn ∈ J(R) for some n ≥ 1 \. This commonly extends the notions of \it fine and \it generalized fine rings defined, respectively, by Călugăreanu-Lam (J. Algebra & Appl., 2016) and Zhou (J. Algebra & Appl., 2022). Specifically, we prove that this class is closed under full matrix rings of any size, as well as we completely characterize when group rings over locally finite groups are generalized √(J)-fine. We also show that every such ring is 2-clean, thus properly placing it between generalized fine rings and 2-clean rings. Several examples are also provided to illustrate the complicated behavior of the introduced concept and its numerous boundaries.