Nil-Clean Rings with Involution

A [Formula: see text]-ring [Formula: see text] is called a nil [Formula: see text]-clean ring if every element of [Formula: see text] is a sum of a projection and a nilpotent. Nil [Formula: see text]-clean rings are the [Formula: see text]-version of nil-clean rings introduced by Diesl. This paper is about the nil [Formula: see text]-clean property of rings with emphasis on matrix rings. We show that a [Formula: see text]-ring [Formula: see text] is nil [Formula: see text]-clean if and only if [Formula: see text] is nil and [Formula: see text] is nil [Formula: see text]-clean. For a 2-primal [Formula: see text]-ring [Formula: see text], with the induced involution given by[Formula: see text], the nil [Formula: see text]-clean property of [Formula: see text] is completely reduced to that of [Formula: see text]. Consequently, [Formula: see text] is not a nil [Formula: see text]-clean ring for [Formula: see text], and [Formula: see text] is a nil [Formula: see text]-clean ring if and only if [Formula: see text] is nil, [Formula: see text]is a Boolean ring and [Formula: see text] for all [Formula: see text].

Graded Modules as a Clean Comodule

In ring and module theory, the cleanness property is well established. If any element of R can be expressed as the sum of an idempotent and a unit, then R is said to be a clean ring. Moreover, an R-module M is clean if the endomorphism ring of M is clean. We study the cleanness concept of coalgebra and comodules as a dualization of the cleanness in rings and modules. Let C be an R-coalgebra and M be a C-comodule. Since the endomorphism of C-comodule M is a ring, M is called a clean C-comodule if the ring of C-comodule endomorphisms of M is clean. In Brzezi´nski and Wisbauer (2003), the group ring R[G] is an R-coalgebra. Consider M as an R[G]-comodule. In this paper, we have investigated some sucient conditions to make M a clean R[G]-comodule, and have shown that every G-graded module M is a clean R[G]-comodule if M is a clean R-module.

Semicommutativity of rings by the way of idempotents

In this paper, we focus on the semicommutative property of rings via idempotent elements. In this direction, we introduce a class of rings, so-called right e-semicommutative rings. The notion of right e-semicommutative rings generalizes those of semicommutative rings, e-symmetric rings and right e-reduced rings. We present examples of right e-semicommutative rings that are neither semicommutative nor e-symmetric nor right e-reduced. Some extensions of rings such as Dorroh extensions and some subrings of matrix rings are investigated in terms of right e-semicommutativity. We prove that if R is a right e-semicommutative clean ring, then the corner ring eRe is clean.

The group ring ℤ(p)Cq and Ye’s theorem

We generalize Ye’s Theorem which states that the group ring [Formula: see text] is a semi-clean ring [Y. Ye, Semiclean rings, Comm. Algebra 31 (2003) 5609–5625]. The proof provided here is more efficient; it is less algorithmic but has the feature that the following statement is evident: for distinct primes [Formula: see text], the group ring [Formula: see text] is feebly clean if and only if the order of [Formula: see text] modulo [Formula: see text] is at least [Formula: see text].

Idempotent elements and uniquely clean property of skew monoid rings

Let R be a ring with an endomorphism σ and F ∪ {0} the free monoid generated by U = {u1, ..., ut} with 0 added, and M = F ∪ {0}/(I) where I is the set of certain monomial in U such that Mn = 0, for some n. Then we can form the non-semiprime skew monoid ring R[M; σ]. An element a ∈ R is uniquely strongly clean if a has a unique expression as a = e + u, where e is an idempotent and u is a unit with ea = ae. We show that a σ-compatible ring R is uniquely clean if and only if R[M; σ] is a uniquely clean ring. If R is strongly π-regular and uniquely strongly clean, then R[M; σ] is uniquely strongly clean. It is also shown that idempotents of R[M; σ] (and hence the ring R[x; σ]=(xn)) are conjugate to idempotents of R and we apply this to show that R[M; σ] over a projective-free ring R is projective-free. It is also proved that if R is semi-abelian and σ(e) = e for each idempotent e ∈ R, then R[M; σ] is a semi-abelian ring.

Rings in which elements are sums of tripotents and nilpotents

A ring [Formula: see text] is strongly 2-nil-clean if every element in [Formula: see text] is the sum of a tripotent and a nilpotent that commute. We prove that a ring [Formula: see text] is strongly 2-nil-clean if and only if [Formula: see text] is a strongly feebly clean 2-UU ring if and only if [Formula: see text] is an exchange 2-UU ring. Furthermore, we characterize strongly 2-nil-clean ring via involutions. We show that a ring [Formula: see text] is strongly 2-nil-clean if and only if every element in [Formula: see text] is the sum of an idempotent, an involution and a nilpotent that commute.

On γ-Clean Rings

In this paper we introduce the concept of -clean ring and we discuss some relations between - clean ring and other rings with explaining by some examples. Also, we give some basic properties of it.

Conjugate (nil) clean rings and Köthe’s problem

Question 3 of [3] asks whether the matrix ring [Formula: see text] is nil clean, for any nil clean ring [Formula: see text]. It is shown that, positive answer to this question is equivalent to positive solution for Köthe’s problem in the class of algebras over the field [Formula: see text]. Other equivalent problems are also discussed. The classes of conjugate clean and conjugate nil clean rings, which lie strictly between uniquely (nil) clean and (nil) clean rings are introduced and investigated.

Rings in which the power of every element is the sum of an idempotent and a unit

A ring R is uniquely ?-clean if the power of every element can be uniquely written as the sum of an idempotent and a unit. We prove that a ring R is uniquely ?-clean if and only if for any a ? R, there exists an integer m and a central idempotent e ? R such that am ? e ? J(R), if and only if R is Abelian; idempotents lift modulo J(R); and R/P is torsion for all prime ideals P ? J(R). Finally, we completely determine when a uniquely ?-clean ring has nil Jacobson radical.