A ring R is a UU ring if every unit is unipotent, or equivalently if every unit is a sum of a nilpotent and an idempotent that commute. These rings have been investigated in Cȃlugȃreanu [3] and in Danchev and Lam [7]. In this paper, two generalizations of UU rings are discussed. We study rings for which every unit is a sum of a nilpotent and an idempotent, and rings for which every unit is a sum of a nilpotent and two idempotents that commute with one another.1 2 KARIMI-MANSOUB, KOSAN, ZHOU nil-clean. Here we are motivated to study rings whose units are nil-clean. These rings will be called UNC rings.In section 2, we first prove several basic properties of UNC rings. Especially it is proved that every semilocal UNC ring is nil-clean. This can be seen as a partial answer to the question of Danchev and Lam. We next show that the matrix ring over a commutative ring R is a UNC ring if and only if R/J(R) is Boolean with J(R) nil. As a consequence, the matrix ring over a UNC ring need not be a UNC ring. We also discuss when a group ring is a UNC (UU) ring. In the last part of this section, it is shown that UU rings are exactly those rings whose units are uniquely nil-clean. As the main result in this section, it is proved that a ring R is strongly nil-clean if and only if R is a semipotent UNC ring.As another natural generalization of UU rings, in section 3 we determine the rings for which every unit is a sum of a nilpotent and two idempotents that commute with one another. We also deal with a special case where every unit of the ring is a sum of two commuting idempotents. These conditions can be compared with the so-called strongly 2nil-clean rings introduced by Chen and Sheibani [4], and the rings for which every element is a sum of two commuting idempotents, studied by Hirano and Tominaga in [11].We write M n (R), T n (R) and R[t] for the n × n matrix ring, the n × n upper triangular matrix ring, and the polynomial ring over R, respectively. For an endomorphism σ of a ring R, let R[t; σ] denote the ring of left skew power series over R. Thus, elements of R[t; σ] are polynomials in t with coefficients in R written on the left, subject to the relation tr = σ(r)t for all r ∈ R. The group ring of a group G over a ring R is denoted by RG. Units being nil-clean2.1. Basic properties. We present various properties of the rings whose units are nil-clean, and prove that every semilocal ring whose units are nil-clean is a nil-clean ring.Definition 2.1. A ring R is called a UNC ring if every unit of R is nil-clean.Every nil-clean ring is a UNC ring. A ring R is called a UU ring if U (R) = 1 + Nil(R) (see [3]).

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