Rings Whose Invertible Elements Are Weakly Nil-Clean

Abstract

We study those rings in which all invertible elements are weakly nil-clean, calling them UWNC rings. This somewhat extends results due to Karimi-Mansoub et al. in Contemp. Math. (2018), where rings in which all invertible elements are nil-clean were considered abbreviating them as UNC rings. Specifically, our main achievements are that the triangular matrix ring Tn(R) over a ring R is UWNC precisely when R is UNC. Besides, the notions UWNC and UNC do coincide when 2 ∈ J(R). We also describe UWNC 2-primal rings R by proving that R is a ring with J(R) = Nil(R) such that U(R) = ±1 + Nil(R). In particular, the polynomial ring R[x] over some arbitrary variable x is UWNC exactly when R is UWNC. Likewise, we furthermore apply the obtained results to group rings showing that if G is a locally finite p-group and R is a UWNC ring such that the prime p is a nilpotent in R, then RG is too a UWNC ring. Some other relevant assertions are proved in the present direction as well.