Two elements $a,b$ in a ring $R$ form a right coprime pair, written $\langle a,b\rangle$, if $aR+bR=R$. Right coprime pairs have shown to be quite useful in the study of left cotorsion or exchange rings. In this paper, we define the class of strongly right exchange rings in terms of descending chains of them. We show that they are semiregular and that this class of rings contains left injective, left pure-injective, left cotorsion, local and left continuous rings. This allows us to give a unified study of all these classes of rings in terms of the behaviour of descending chains of right coprime pairs.
We show that a suitable ring with a ``nice'' topology, in which convergent limits of units are units, is an \aleph_0-exchange ring. We generalize the argument to show that a semi-regular ring, R, with a ``nice'' topology, is a full exchange ring. Putting these results in the language of modules, we show that a cohopfian module with finite exchange has countable exchange. Also, all modules with Dedekind-finite, semi-regular endomorphism rings are full exchange modules.
In this paper we discuss several constructions that lead to new examples of nil-clean, clean, and exchange rings. A characterization of the idempotents in the algebra defined by a 2-cocycle is given and used to prove some of the algebra's properties (the infinitesimal deformation case). From infinitesimal deformations we go to full deformations and prove that any formal deformation of a clean (exchange) ring is itself clean (exchange). Examples of nil-clean, clean, and exchange rings arising from poset algebras are also discussed.
An exchange ring $R$ is separative provided that for all finitely generated projective right $R$-modules $A$ and $B$, $A\oplus A\cong A\oplus B\cong B\oplus B\Longrightarrow A\cong B$. Let $R$ be a separative exchange ring in which $2$ is invertible, and let $a-a^3\in R$ be regular. We prove, in this note, that $a\in R$ is unit-regular if $R(1-a^2)R=Rr(a)={\ell}(a)$. An element $a$ in a ring $R$ is special clean if there exists an idempotent $e\in R$ such that $a-e\in R$ is a unit and $aR\bigcap eR=0$. Furthermore, we prove that $a\in R$ is special clean if $aR/ar(a^2), R/\big(aR+r(a)\big)$ ar…
Let $R$ be an exchange ring. We prove that the relative elementary subgroups $E_n(R,I)$ are normal in the general linear group $GL_n(R)$ if $n\geq 1$ and that the standard commutator formula $E_n(R,I)=[E_n(R),E_n(R,I)]=[E_n(R),C_n(R,I)]$ holds if $n\geq 3$. Moreover, we classify the subgroups of $GL_n(R)$ that are normalised by the elementary subgroup $E_n(R)$ in the case $n\geq 3$.
Let $R$ be an associative ring with identity, $C(R)$ denote the center of $R$, and $g(x)$ be a polynomial in the polynomial ring $C(R)[x]$. $R$ is called strongly $g(x)$-clean if every element $r \in R$ can be written as $r=s+u$ with $g(s)=0$, $u$ a unit of $R$, and $su=us$. The relation between strongly $g(x)$-clean rings and strongly clean rings is determined, some general properties of strongly $g(x)$-clean rings are given, and strongly $g(x)$-clean rings generated by units are discussed.
A ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute. By {\rm SRC} factorization, Borooah, Diesl, and Dorsey \cite{BDD051} completely determined when ${\mathbb M}_n(R)$ over a commutative local ring $R$ is strongly clean. We generalize the notion of {\rm SRC} factorization to commutative rings, prove that commutative $n$-{\rm SRC} rings $(n\ge 2)$ are precisely the commutative local rings over which ${\mathbb M}_n(R)$ is strongly clean, and characterize strong cleanness of matrices over commutative projective-free rings having {\rm ULP…
Let $G$ be a group and let $S=\bigoplus_{g \in G} S_g$ be a $G$-graded ring. Given a normal subgroup $N$ of $G$, there is a naturally induced $G/N$-grading of $S$. It is well-known that if $S$ is strongly $G$-graded, then the induced $G/N$-grading is strong for any $N$. The class of epsilon-strongly graded rings was recently introduced by Nystedt, Öinert and Pinedo as a generalization of unital strongly graded rings. We give an example of an epsilon-strongly graded partial skew group ring such that the induced quotient group grading is not epsilon-strong. Moreover, we give necessary and suffic…
A *-ring $R$ is called a strongly nil-*-clean ring if every element of $R$ is the sum of a projection and a nilpotent element that commute with each other. In this article, we show that $R$ is a strongly nil-*-clean ring if and only if every idempotent in $R$ is a projection, $R$ is periodic, and $R/J(R)$ is Boolean. For any commutative *-ring $R$, we prove that the algebraic extension $R[i]$ where $i^2=μi+η$ for some $μ,η\in R$ is strongly nil-*-clean if and only if $R$ is strongly nil-*-clean and $μη$ is nilpotent. The relationships between Boolean *-rings and strongly nil-*-clean rings are …
An element $a$ in a ring $R$ is strongly J-clean if it is the sum of an idempotent and an element in the Jacobson radical that commutes. We characterize the strongly J-clean $2\times 2$ matrices over 2-projective-free non-commutative rings.
The literature about strongly clean matrices over commutative rings is quite extensive. The sharpest results are about matrices over commutative local rings, for example those by Borooah, Diesl and Dorsey. The purpose of this note is to show that, using Pierce sheaf techniques, many of the known results about matrices over commutative local rings can be extended to those over commutative clean rings in general.
An element $a\in R$ is provided that there exists an idempotent $e\in R$ such that $a-e\in U(R), ae=ea$ and $eae\in J(eRe)$. In this article, we investigate strongly rad-clean matrices over a commutative local ring. We completely determine when a $2\times 2$ matrix over a commutative local ring is strongly rad-clean.
Many authors have investigated the behavior of strong cleanness under certain ring extensions. In this note, we prove that if $R$ is a ring which is complete with respect to an ideal $I$ and if $x$ is an element of $R$ whose image in $R/I$ is strongly $π$-regular, then $x$ is strongly clean in $R$.
We introduce the class of epsilon-strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products. We determine precisely when an epsilon-strongly graded ring is separable over its principal component. Thereby, we simultaneously generalize a result for strongly group graded rings by Nastasescu, Van den Bergh and Van Oystaeyen, and a result for unital partial crossed products by Bagio, Lazzarin and Paques. We also show that the class of unital partial crossed products appear in the class of epsilon-strongly graded…
We investigate and characterize several kinds of elements such as units, idempotents, von Neumann regular, $π$-regular and clean elements for skew PBW extensions over weak compatible rings. We also study the notions of Gelfand and Harmonic rings for these families of algebras. The results presented here extend those corresponding in the literature for commutative and noncommutative rings of polynomial type.
Symmetric rings were introduced by Lambek to extend usual commutative ideal theory in noncommutative rings. In this paper, we study symmetric rings over which Ore extensions are symmetric. A ring R is called strongly σ-symmetric if the skew polynomial ring R[x;σ] is symmetric. We consider some properties and extensions of strongly σ-symmetric rings. Then we show the relationship between strongly σ-symmetric rings and other classes of rings. We next argue the polynomial extensions over strongly σ-symmetric rings. Moreover, we prove that if R is a σ-rigid ring, then R[x]/(x_n) is a strongly σ-sy…
We introduce and investigate the so-called D-regularly nil clean rings by showing that these rings are, in fact, a non-trivial generalization of the classical von Neumann regular rings and of the strongly $π$-regular rings. Some other close relationships with certain well-known classes of rings such as $π$-regular rings, exchange rings, clean rings, nil-clean rings, etc., are also demonstrated. These results somewhat supply a recent publication of the author in Turk. J. Math. (2019) as well as they somewhat expand the important role of the two examples of nil-clean rings obtained by Ster in Li…
We describe isomorphisms between strongly triangular matrix rings that were defined earlier in Berkenmeier et al. (2000) as ones having a complete set of triangulating idempotents, and we show that the so-called triangulating idempotents behave analogously to idempotents in semiperfect rings. This study yields also a way to compute theoretically the automorphism groups of such rings in terms of corresponding automorphism groups of certain subrings and bimodules involved in their structure.
A ring with an involution * is called strongly $J$-*-clean if every element is a sum of a projection and an element of the Jacobson radical that commute. In this article, we prove several results characterizing this class of rings. It is shown that a *-ring $R$ is strongly $J$-*-clean, if and only if $R$ is uniquely clean and strongly *-clean, if and only if $R$ is uniquely strongly *-clean, that is, for any $a\in R$, there exists a unique projection $e\in R$ such that $a-e$ is invertible and $ae=ea$.
An element of a ring $R$ is strongly $P$-clean provided that it can be written as the sum of an idempotent and a strongly nilpotent element that commute. A ring $R$ is strongly $P$-clean in case each of its elements is strongly $P$-clean. We investigate, in this article, the necessary and sufficient conditions under which a ring $R$ is strongly $P$-clean. Many characterizations of such rings are obtained. The criteria on strong $P$-cleanness of $2\times 2$ matrices over commutative local rings are also determined.
$\textbf{Theorem 1.3.}$ For a given ring $A$ with right Goldie radical $G(A_A)$, the following conditions are equivalent. $\textbf{1)}$ Every non-singular right $A$-module $X$ which is is injective with respect to some essential right ideal of the ring $A$ is an injective module. $\textbf{2)}$ $A/G(A_A)$ is a right strongly semiprime ring. $\textbf{Theorem 1.4.}$ For a given ring $A$, the following conditions are equivalent. $\textbf{1)}$ $A$ is a right strongly semiprime ring. $\textbf{2)}$ Every right $A$-module which is injective with respect to some essential right ideal of the ring $A$, i…
Rings Whose Non-Invertible Elements Are Uniquely Strongly Clean
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Shahed University（沙赫德大学）；Tarbiat Modares University（塔比尔·莫达雷斯大学）
In this paper we define and study quasipolar general rings (with or without identity) and extend many of the basic results to the wider class. We obtain some new characterizations of quasipolar and strongly $π$-regular elements by using quasipolar general rings. We see that quasipolar general rings lies between strongly $π$-regular and strongly clean general rings. Consequently, we prove that $R$ is pseudopolar if and only if $R$ is strongly $π$-rad clean and quasipolar.
A ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey \cite{BDD05a} completely characterized the commutative local rings $R$ for which ${\mathbb M}_n(R)$ is strongly clean. For a general local ring $R$ and $n>1$, however, it is unknown when the matrix ring ${\mathbb M}_n(R)$ is strongly clean. Here we completely determine the local rings $R$ for which ${\mathbb M}_2(R)$ is strongly clean.
A ring $R$ is uniquely (strongly) clean provided that for any $a\in R$ there exists a unique idempotent $e\in R \big(\in comm(a)\big)$ such that $a-e\in U(R)$. Let $R$ be a uniquely bleached ring. We prove, in this note, that $R$ is uniquely clean if and only if $R$ is abelian, and $T_n(R)$ is uniquely strongly clean for all $n\geq 1$, if and only if $R$ is abelian, $T_n(R)$ is uniquely strongly clean for some $n\geq 1$. In the commutative case, the more explicit results are obtained. These also generalize the main theorems in [6] and [7], and provide many new class of such rings.
A ring $R$ is strongly clean provided that every element in $R$ is the sum of an idempotent and a unit that commutate. Let $T_n(R,σ)$ be the skew triangular matrix ring over a local ring $R$ where $σ$ is an endomorphism of $R$. We show that $T_2(R,σ)$ is strongly clean if and only if for any $a\in 1+J(R), b\in J(R)$, $l_a-r_{σ(b)}: R\to R$ is surjective. Further, $T_3(R,σ)$ is strongly clean if $l_{a}-r_{σ(b)}, l_{a}-r_{σ^2(b)}$ and $l_{b}-r_{σ(a)}$ are surjective for any $a\in U(R),b\in J(R)$. The necessary condition for $T_3(R,σ)$ to be strongly clean is also obtained.
We introduce a generalisation of Condition (K) to finitely separated graphs and show that it is equivalent to essential freeness of the associated partial action as well as the exchange property of any of the associated tame algebras. As a consequence, we can show that any tame separated graph algebra with the exchange property is separative.
A commutative ring $R$ is projective free provided that every finitely generated $R$-module is free. An element in a ring is strongly clean provided that it is the sum of an idempotent and a unit that commutates. Let $R$ be a projective-free ring, and let $h\in R[t]$ be a monic polynomial of degree $n$. We prove, in this article, that every $φ\in M_n(R)$ with characteristic polynomial $h$ is strongly clean, if and only if the companion matrix $C_h$ of $h$ is strongly clean, if and only if there exists a factorization $h=h_0h_1$ such that $h_0\in {\Bbb S}_0, h_1\in {\Bbb S}_1$ and $(h_0,h_1)=1$…
An element of a ring $R$ is called strongly $J^{\#}$-clean provided that it can be written as the sum of an idempotent and an element in $J^{\#}(R)$ that commute. We characterize, in this article, the strongly $J^{\#}$-cleanness of matrices over projective-free rings. These extend many known results on strongly clean matrices over commutative local rings.
Let $G$ be a group with neutral element $e$ and let $S=\bigoplus_{g \in G}S_g$ be a $G$-graded ring. A necessary condition for $S$ to be noetherian is that the principal component $S_e$ is noetherian. The following partial converse is well-known: If $S$ is strongly-graded and $G$ is a polycyclic-by-finite group, then $S_e$ being noetherian implies that $S$ is noetherian. We will generalize the noetherianity result to the recently introduced class of epsilon-strongly graded rings. We will also provide results on the artinianity of epsilon-strongly graded rings. As our main application we obtain…
A ring R is a strongly 2-nil-clean if every element in R is the sum of two idempotents and a nilpotent that commute. A ring R is feebly clean if every element in R is the sum of two orthogonal idempotents and a unit. In this paper, strongly 2-nil-clean rings are studied with an emphasis on their relations with feebly clean rings. This work shows new interesting connections between strongly 2-nil-clean rings and weakly exchange rings
An element $a$ of a ring $R$ is called \emph{strongly $J$-clean} provided that there exists an idempotent $e\in R$ such that $a-e\in J(R)$ and $ae=ea$. A ring $R$ is \emph{strongly $J$-clean} in case every element in $R$ is strongly $J$-clean. In this paper, we investigate strong $J$-cleanness of $M_2(R;s)$ for a local ring $R$ and $s\in R$. We determine the conditions under which elements of $M_2(R;s)$ are strongly $J$-clean.
General cryptographic schemes are presented where keys can be one-time or ephemeral. Processes for key exchange are derived. Public key cryptographic schemes based on the new systems are easily established. Authentication and signature schemes are implemented. The schemes are an advance on group ring techniques and are easily implemented but highly secure. They may be integrated with error-correcting coding schemes so that encryption/coding and decryption/decoding may be done simultaneously.
We consider some existing results regarding rings for which the classes of torsion-free and non-singular right modules coincide. Here, a right $R$-module $M$ is non-singular if $xI$ is nonzero for every nonzero $x \in M$ and every essential right ideal $I$ of $R$, and a right $R$-module $M$ is torsion-free if $Tor_{1}^{R}(M,R / Rr)=0$ for every $r \in R$. In particular, we consider a ring $R$ for which the classes of torsion-free and non-singular right $S$-modules coincide for every ring $S$ Morita-equivalent to $R$. We make use of these results, as well as the existence of a Morita-equivalenc…
We determine the commutant of homogeneous subrings in strongly groupoid graded rings in terms of an action on the ring induced by the grading. Thereby we generalize a classical result of Miyashita from the group graded case to the groupoid graded situation. In the end of the article we exemplify this result. To this end, we show, by an explicit construction, that given a finite groupoid $G$, equipped with a nonidentity morphism $t : d(t) \to c(t)$, there is a strongly $G$-graded ring $R$ with the properties that each $R_s$, for $s \in G$, is nonzero and $R_t$ is a nonfree left $R_{c(t)}$-modul…
We introduce the class of partially invertible modules and show that it is an inverse category which we call the Picard inverse category. We use this category to generalize the classical construction of crossed products to, what we call, generalized epsilon-crossed products and show that these coincide with the class of epsilon-strongly groupoid graded rings. We then use generalized epsilon-crossed groupoid products to obtain a generalization, from the group graded situation to the groupoid graded case, of the bijection from a certain second cohomology group, defined by the grading and the fun…
A $*$-ring $R$ is called (strongly) $*$-clean if every element of $R$ is the sum of a projection and a unit (which commute with each other). In this note, some properties of $*$-clean rings are considered. In particular, a new class of $*$-clean rings which called strongly $π$-$*$-regular are introduced. It is shown that $R$ is strongly $π$-$*$-regular if and only if $R$ is $π$-regular and every idempotent of $R$ is a projection if and only if $R/J(R)$ is strongly regular with $J(R)$ nil, and every idempotent of $R/J(R)$ is lifted to a central projection of $R.$ In addition, the stable range c…
In this article, we introduce the weak ideal-Armendariz ring which combines Armendariz ring and weakly semicommutative properties of rings. In fact, it is a generalisation of an ideal-Armendariz ring. We investigate some properties of weak ideal Armendariz rings and prove that R is a weak ideal-Armendariz ring if and only if R[x] is weak ideal-Armendariz ring. Also, we generalise weak ideal-Armendariz as strongly nil-IFP and a number of properties are discussed which distinguishes it from other existing structures. We prove that if I is a semicommutative ideal of a ring R and R/I is a strongly…
In this paper, strongly clean ring defined by W. K. Nicholson in 1999 has been generalized to n-strongly clean, Σ-strongly clean and with the help of example it has been shown that there exists a ring, which is n-strongly clean and Σ-strongly clean but not strongly clean. It has been shown that for a commutative ring R formal power series R[(x)] of R is n-strongly clean if and only if R is n- strongly clean. We also discussed the structure of homomorphic image of n- strongly clean and direct product of n- strongly clean rings. It has also been shown that for any commutative ring R, the polynom…
A ring $R$ is (strongly) 2-nil-clean if every element in $R$ is the sum of two idempotents and a nilpotent (that commute). Fundamental properties of such rings are discussed. Let $R$ be a 2-primal ring. If $R$ is strongly 2-nil-clean, we show that $M_n(R)$ is 2-nil-clean for all $n\in {\Bbb N}$. We also prove that the matrix ring is 2-nil-clean for a strongly 2-nil-clean ring of bounded index. These provide many classes of rings over which every matrix is the sum of two idempotents and a nilpotent.
Let $R$ be a commutative local ring. It is proved that $R$ is Henselian if and only if each $R$-algebra which is a direct limit of module finite $R$-algebras is strongly clean. So, the matrix ring $\mathbb{M}_n(R)$ is strongly clean for each integer $n>0$ if $R$ is Henselian and we show that the converse holds if either the residue class field of $R$ is algebraically closed or $R$ is an integrally closed domain or $R$ is a valuation ring. It is also shown that each $R$-algebra which is locally a direct limit of module-finite algebras, is strongly clean if $R$ is a $π$-regular commutative ring.
In this work we study the classes of epsilon and nearly epsilon-strongly graded rings by a group $G$. In particular, we extend Dade's theorem to the realm of nearly epsilon-strongly graded rings. Moreover, we introduce the category SIM$S-$gr of symmetrically graded modules and use it to present a new characterization of strongly graded rings. A functorial approach is used to obtain a characterization of epsilon-strongly graded rings. Finally, we determine conditions for which an epsilon-strongly graded ring can be written as a direct sum of strongly graded rings and a trivially graded ring.
The assignment (nonstable K_0-theory), that to a ring R associates the monoid V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor: (1) There is no functor F, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that VF is equivalent to the identity. (2) There is no functor F, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp.,…
For any (unital) exchange ring $R$ whose finitely generated projective modules satisfy the separative cancellation property ($A\oplus A\cong A\oplus B\cong B\oplus B$ implies $A\cong B$), it is shown that all invertible square matrices over $R$ can be diagonalized by elementary row and column operations. Consequently, the natural homomorphism $GL_1(R) \to K_1(R)$ is surjective. In combination with a result of Huaxin Lin, it follows that for any separative, unital C*-algebra $A$ with real rank zero, the topological $K_1(A)$ is naturally isomorphic to the unitary group $U(A)$ modulo the connecte…
The algebraic Cuntz-Pimsner rings are naturally $\mathbb{Z}$-graded rings that generalize both Leavitt path algebras and unperforated $\mathbb{Z}$-graded Steinberg algebras. We classify strongly, epsilon-strongly and nearly epsilon-strongly graded algebraic Cuntz-Pimsner rings up to graded isomorphism. As an application, we characterize noetherian and artinian fractional skew monoid rings by a single corner automorphism.
In this paper we present a new primitive for a key exchange protocol based on multivariate non-commutative polynomial rings, analogous to the classic Diffie-Hellman method. Our technique extends the proposed scheme of Boucher et al. from 2010. Their method was broken by Dubois and Kammerer in 2011, who exploited the Euclidean domain structure of the chosen ring. However, our proposal is immune against such attacks, without losing the advantages of non-commutative polynomial rings as outlined by Boucher et al. Moreover, our extension is not restricted to any particular ring, but is designed to …
University of Johannesburg（约翰内斯堡大学）；National Institute for Theoretical and Computational Sciences (NITheCS)（理论与计算科学国家研究所）；University of Cape Town（开普敦大学）
