Narrow Results

We investigate the transfer of the notion of feebly clean along with related concepts such as feebly $J$-clean, and feebly nil-clean rings, to bi-amalgamated algebras along ideals. For example, we prove that the ring $A{\bowtie }^{f,g}(J,J^{\prime })$ is feebly clean ($J$-clean, nil-clean) under the conditions $J\times J^{\prime }\subseteq Nil(B\times C)$, and one of the rings $A$, $f(A)+J$, and $g(A)+J^{\prime }$ is feebly clean ($J$-clean, nil-clean). Also, we provide new characterizations of feebly $J$-clean and feebly clean rings. All along the paper, we use the results to build examples subject to the mentioned concepts.

A ring $R$ is called clean (respectively, semi-clean) if each element of $R$ can be written as the sum of a unit and an idempotent (respectively, a periodic element). In this paper, we seek the necessary and sufficient conditions under which, for a ring $R$ and a group $G$ with additional conditions, the group ring $RG$ is clean or semi-clean. As a remarkable result, let $R$ be an Abelian clean ring, a reduced ring, or a commutative ring, and let $G$ be a locally nilpotent group, we show that, if $RG$ is semi-clean, then $G$ is locally finite. Also, we show that if $R$ is a semi-local ring whose Jacobson radical is locally nilpotent and $G$ is a locally finite group, then $RG$ is clean. These results generalize some earlier results in the literature.

A ring $R$ is called $m$-clean if each element of $R$ can be expressed as the sum of a unit and an $m$-potent in $R$. We introduce the concept of the $m$-clean index associated with a ring $R$. For any element $a\in R$, we define $E_m(a)=\{e\in R:e^m=e\text{and}a-e\in U(R)\}$, where $U(R)$ denotes the set of units of $R$. Then the $m$-clean index of the ring $R$ is defined as $sup\{|E_m(a)|:a\in R\}$ and it is denoted by $m$-$c$-$ind(R)$. In this paper, we study some properties of this index and characterize the ring of $m$-clean index $1$. We show that the rings with the $m$-clean index 1 are precisely the rings whose $2$-clean index is $1$ and each $m$-potent is idempotent. We establish some connections between $m$-clean index of the ring $\left(\begin{array}{cc}\hfill A\hfill & \hfill M\hfill \\ \hfill O\hfill & \hfill B\hfill \end{array}\right)$ and the underlying rings $A,B$ and $R$-module $M$. We also determine the $m$-clean index of the trivial extension $R\propto M$ and the $m$-clean index of the ideal extension $I(R,V)$ with some associated conditions. Finally, we discuss the amalgamated structure of ring in connection to the $m$-clean index.

Commutative clean rings and related rings have received much recent attention. A ring R is clean if each r ∈ R can be written r = u + e, where u is a unit and e an idempotent. This article deals mostly with the question: When is the classical ring of quotients of a commutative ring clean? After some general results, the article focuses on C(X) to characterize spaces X when Q_cl(X) is clean. Such spaces include cozero complemented, strongly 0-dimensional and more spaces. Along the way, other extensions of rings are studied: directed limits and extensions by idempotents.

A *-ring R is called a *-clean ring if every element of R is the sum of a unit and a projection, and R is called a strongly *-clean ring if every element of R is the sum of a unit and a projection that commute with each other. These concepts were introduced and discussed recently by [L. Vaš, *-Clean rings; some clean and almost clean Baer *-rings and von Neumann algebras, J. Algebra324 (2010) 3388–3400]. Here it is proved that a *-ring R is strongly *-clean if and only if R is an abelian, *-clean ring if and only if R is a clean ring such that every idempotent is a projection. As consequences, various examples of strongly *-clean rings are constructed and, in particular, two questions raised in [L. Vaš, *-Clean rings; some clean and almost clean Baer *-rings and von Neumann algebras, J. Algebra324 (2010) 3388–3400] are answered.

Let R be a commutative ring with identity and let I be an ideal of R. In this paper we study the amalgamated duplication ring R ⋈ I introduced by D'Anna and Fontana. By assuming that (1, i) is a unit in R ⋈ I for every i ∈ I, it is shown that R ⋈ I is uniquely clean (respectively, n-clean, weakly clean, n-good) if and only if R is uniquely clean (respectively, n-clean, weakly clean, n-good). Conditions for R ⋈ I to be almost clean are also discussed.

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C₃, C₄, S₃ and Q₈. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 𝔽C_p is *-clean, where 𝔽 is a field and C_p is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 𝔽. Among other results we also obtain a complete characterization of when RC_n (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring.

In [A. J. Diesl, Classes of strongly clean rings, Ph.D. Dissertation, University of California, Berkely (2006); Nil clean rings, J. Algebra383 (2013) 197–211], a nil clean ring was defined as a ring for which every element is the sum of a nilpotent and an idempotent. In this short paper, we characterize nil clean commutative group rings.

We generalize Ye’s Theorem which states that the group ring ${\mathbb{Z}}_{(p)}[C_3]$ is a semi-clean ring [Y. Ye, Semiclean rings, Comm. Algebra31 (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 $p,q$, the group ring ${\mathbb{Z}}_{(p)}{C}_q$ is feebly clean if and only if the order of $p$ modulo $q$ is at least $\frac{q-1}{2}$.

An element of a ring $R$ is said to be $r$-precious if it can be written as the sum of a von Neumann regular element, an idempotent element and a nilpotent element. If all the elements of a ring $R$ are $r$-precious, then $R$ is called an $r$-precious ring. We study some basic properties of $r$-precious rings. We also characterize von Neumann regular elements in $M_2(R)$ when $R$ is a Euclidean domain and by this argument, we produce elements that are $r$-precious but either not $r$-clean or not precious.

A ring $R$ is said to be clean if each element of $R$ can be written as the sum of a unit and an idempotent. $R$ is said to be weakly clean if each element of $R$ is either a sum or a difference of a unit and an idempotent, and $R$ is said to be feebly clean if every element $r$ can be written as $r=u+e_1-e_2$, where $u$ is a unit and $e_1,e_2$ are orthogonal idempotents. Clearly, clean rings are weakly clean rings and both of them are feebly clean. In a recent paper (J. Algebra Appl.17 (2018) 1850111 (5 pp.)), McGoven characterized when the group ring ${\mathbb{Z}}_{(p)}[C_q]$ is weakly clean and feebly clean, where $p,q$ are distinct primes. In this paper, we consider a more general setting. Let $K$ be an algebraic number field, ${𝒪}$ its ring of integers, $𝔭\subset {𝒪}$ a nonzero prime ideal and ${{𝒪}}_{𝔭}$ the localization of ${𝒪}$ at $𝔭$. We investigate when the group ring ${{𝒪}}_{𝔭}[G]$ is weakly clean and feebly clean, where $G$ is a finite abelian group, and establish an explicit characterization for such a group ring to be weakly clean and feebly clean for the case when $K=\mathbb{Q}({\zeta }_n)$ is a cyclotomic field or $K=\mathbb{Q}(\sqrt{d})$ is a quadratic field.

For $n\ge 2$ and for a ring $R,$ the notation $P_n(R)$ means that $r^n-r$ is nilpotent for all $r\in R$. In this paper, rings $R$ for which $P_n(R)$ holds are completely characterized for any integers $n\ge 2$. This answers a question which was raised in [T. Kosan, Y. Zhou, T. Yildirim, Rings with $x^n-x$ nilpotent, J. Algebra Appl.19(4) (2020) 2050065].

Various authors have been generalizing some unital ring properties to nonunital rings. We consider properties related to cancellation of modules (being unit-regular, having stable range one, being directly finite, exchange, or clean) and their “local” versions. We explore their relationships and extend the defined concepts to graded rings. With graded clean and graded exchange rings suitably defined, we study how these properties behave under the formation of graded matrix rings. We exhibit properties of a graph E which are equivalent to the unital Leavitt path algebra $L_K(E)$ being graded clean. We also exhibit some graph properties which are necessary and some which are sufficient for $L_K(E)$ to be graded exchange.

Let R be a commutative ring with nonzero identity. In this paper, we study the von Neumann regular elements of R. We also study the idempotent elements, π-regular elements, the von Neumann local elements, and the clean elements of R. Finally, we investigate the subgraphs of the zero-divisor graph Γ(R) of R induced by the above elements.

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection. It is obvious that ∗-clean rings are clean. Vaš asked whether there exists a clean ring with involution that is not ∗-clean. In this paper, we investigate when a group ring RG is ∗-clean, where ∗ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ∗-clean, where R is a commutative local ring and G is one of the groups C₃, C₄, S₃ and Q₈. As a consequence, we provide many examples of group rings which are clean but not ∗-clean.

Let R be a ring with an endomorphism σ. We show that the clean property and various Armendariz-type properties of R are inherited by the skew matrix rings S(R,n,σ) and T(R,n,σ). They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which inherit various interesting properties of rings.