P-clean rings

Throughout this paper R denotes an associative ring with identity and all modules are unitary. We use the symbol U(R) to denote the group of units of R and Id(R) the set of idempotents of R, Un(R) the set of elements which are the sum of n units of R, UΣ(R) the set of elements each of which is the sum of finitely many units in R, RE(R) (URE(R)) the set of regular (unit regular) elements of R, and Peri(R) the set of periodic elements of R. The Jacobson radical and the prime radical of R are denoted by J(R) and Nil∗(R), respectively. Following Han and Nicholson [4], an element x of a ring R is called clean if x = e+u where e ∈ Id(R) and u∈U(R). A ring R is clean if every element of R is clean. This notion was first introduced by Nicholson [5] as early as 1977 in his study of lifting idempotents and exchange rings. Since then, a great deal is known about clean rings and their generalizations (cf. [1–9]). According to Ye [9], a ring R is called semiclean if every element of R has the form x = f + u, where u ∈ U(R) and f is periodic, that is, f p = f q for two different positive integers p and q . In [8], an element x of a ring R is called n-clean if x = e+u1 + ···+un where e ∈ Id(R), ui ∈ U(R), and n is a positive integer. The ring R is called n-clean if every element of R is n-clean for some fixed positive integer n. While R is called Σ-clean, if the n is a positive integer depending on x. Also Zhang and Tong in [10] defined R to be G-clean, if each x ∈ R has the form x = a+u where a is unit regular and u∈U(R). Motivated by the results of Han and Nicholson [4] on clean rings, Ye [9] on semiclean rings, Xiao and Tong [8] on n-clean rings and Σ-clean rings, and Zhang and Tong [10] on G-clean rings, in this paper we unify the structures of various clean rings by introducing