Exchange Rings Research Articles

Quasi-exchange rings

An element a in a ring R is called a quasi-idempotent if a 2 = ka for some central unit k of R. A ring R is called quasi-exchange if for any element x of R, there exists a ∈ R with a 2 = ka ∈ Rx and k − a ∈ R ( k − x ) , where k is a central unit of R. Quasi-exchange rings are a generalization of exchange rings and quasi-clean rings. This paper studies the properties of quasi-exchange rings and the relationship among quasi-exchange rings, exchange rings and quasi-clean rings. An equation xa − fx = 1 is used to characterize the quasi-exchange ring and we prove that quasi-exchange rings are left-right symmetric.

Cultural Acculturation of Ring Exchange in the Mappettu Ada Tradition from the Perspective of Islamic Law

This research examines the acculturation of ring exchange culture in the mappettu ada tradition carried out by the Cempa community of Pinrang Regency with the problems in the research, namely, First, how is the acculturation of the ring exchange culture in the mappettu tradition in Cempa Pinrang Regency? Second, what is the practice of exchanging rings at the mappettu event in Cempa, Pinrang Regency? Third, what is the review of Islamic law regarding exchanging rings at the mappettu event in Cempa, Pinrang Regency?This research is a type of field research using a normative approach that is descriptive qualitative in nature. The data collection techniques that the author uses are observation and interview techniques.The research results show that: (1) Exchanging rings at the mappettu event is a cultural acculturation process influenced by external factors (Western culture) which is imitated by the Bugis community in Cempa which brings the bride and groom together. (2) In the practice of exchanging rings at the mappettu event in Cempa, Pinrang Regency, namely with the development of modern times, the exchange of rings is carried out like a party by decorating the room, the bride and groom put rings on each other. gold ring directly on the left ring finger. (3) Judging from Islamic law, the practice of exchanging rings is not in accordance with the Shari'a where in practice there are elements between the prospective groom and the prospective bride touching each other when pairing the rings. The use of gold on men is also prohibited in Islamic law, although only a little, and is permitted for women as explained in the hadith. Keywords: Acculturation,Exchange Rings, Islamic Law

On some generalizations of abelian rings

Semiabelian rings, defined by the property that each of their idempotents is either left semicentral or right semicentral, are one among several natural generalizations of abelian rings. In this semi-expository paper, we review a number of interesting properties of semiabelian (and other closely allied) rings that are so far well known only for abelian rings. For instance, semiabelian rings [Formula: see text] are always “J-abelian” in the sense that each idempotent of [Formula: see text] maps onto a central idempotent in [Formula: see text]. On the other hand, J-abelian rings turn out to be precisely the “strongly perspective rings” as well as the “strongly IC rings”, and the von Neumann regular elements in such rings are automatically strongly regular and are closed under taking [Formula: see text]-th powers. In addition, all J-abelian exchange rings have idempotent stable range one, and are in particular clean (although not necessarily strongly clean) rings.

Strong exchange rings

Strong exchange rings

An introduction to q-central idempotents and q-abelian rings

A ring idempotent is said to be quarter-central (or q-central for short) if . If all idempotents in a ring are q-central, is said to be a quarter-abelian ring (or more simply, a q-abelian ring). In this paper, we show that such a ring is characterized by the property that for all idempotents , where denotes the additive commutator . For any nonzero ring and any integer , we show that the ring of upper triangular matrices over is not q-abelian. On the other hand, is q-abelian iff is abelian (in the classical sense that all idempotents are central in ). A final section of this paper relates q-central idempotents to the notion of exchange rings and the study of regular, unit-regular, and strongly regular elements in arbitrary rings. From the viewpoint of currently prevailing generalizations of abelian rings in the literature, q-abelian rings are situated between the class of semiabelian rings and the class of strongly IC rings.

C4-modules with the exchange property

We prove that if M is a C4-module (D4-module) with the restricted ascending (descending) chain condition on direct summands, then M satisfies the finite exchange iff M is clean, iff M satisfies the full exchange. This result extends an earlier one by the authors on summand-square-free (summand-dual-square-free) modules. Moreover, as an immediate application, it follows that every exchange ring with the restricted DCC on direct summands is clean. The later result is a non-trivial and simultaneous extension of two fundamental results on exchange rings, the first of which is due to Nicholson and says that Abelian exchange rings are clean, and the second is due to Camillo and Yu and says that exchange rings with no infinite sets of orthogonal idempotents are clean.

ON RINGS IN WHICH ALL UNITS CAN BE PRESENTED IN THE FORM 1+eR(1-e)

The aim of the present paper is to characterize associative rings R with unity in which 1+eR(1-e) in terms of some important class of rings in the literature (for example, NR-rings, UU-rings, UJ-rings, UR-rings, exchange rings, 2-primal rings), where e2=e∈ R and U(R) is the set of units of R.

Uniquely exchange rings

An associative ring with unity is called exchange if every element is exchange, i.e., there exists an idempotent e ? aR such that 1?e ? (1?a)R; if this representation is unique for every element, we call the ring uniquely exchange. We give a complete description of uniquely exchange rings.

Cancellation properties of graded and nonunital rings. Graded clean and graded exchange Leavitt path algebras

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 [Formula: see text] being graded clean. We also exhibit some graph properties which are necessary and some which are sufficient for [Formula: see text] to be graded exchange.

A symmetric generalization of $$\pi $$-regular rings

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 $$\pi $$ -regular rings (in particular, of the von Neumann regular rings and of the strongly $$\pi $$ -regular rings). Some other close relationships with certain well-known classes of rings such as 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 Linear Algebra Appl (2018). Likewise, the obtained symmetrization supports that similar property for exchange rings established by Khurana et al. in Algebras Represent Theory (2015).

Rings additively generated by idempotents and nilpotents

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 article, 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.

On general linear groups over exchange rings

Let R be an exchange ring. We prove that the relative elementary subgroups are normal in the general linear group if and that the standard commutator formula holds if . Moreover, we classify the subgroups of that are normalized by the elementary subgroup in the case .

Boolean lifting property in quantales

In ring theory, the lifting idempotent property (LIP) is related to some important classes of rings: clean rings, exchange rings, local and semilocal rings, Gelfand rings, maximal rings, etc. Inspired by LIP, lifting properties were also defined for other algebraic structures: MV-algebras, BL-algebras, residuated lattices, abelian l-groups, congruence distributive universal algebras, etc. In this paper, we define a lifting property (LP) in commutative coherent integral quantales, structures that are a good abstraction for lattices of ideals, filters and congruences. LP generalizes all the lifting properties existing in the literature. The main tool in the study of LP in a quantale A is the reticulation of A, a bounded distributive lattice whose prime spectrum is homeomorphic to the prime spectrum of A. The principal results of the paper include a characterization theorem for quantales with LP and a characterization theorem for hyperarchimedean quantales.

On a common generalization of symmetric rings and quasi duo rings

Let \(J(R)\) denote the Jacobson radical of a ring \(R\). We call a ring \(R\) as \(J\)-symmetric if for any \(a,b, c\in R, abc=0\) implies \(bac\in J(R)\). It turns out that \(J\)-symmetric rings are a common generalization of left (right) quasi-duo rings and generalized weakly symmetric rings. Various properties of these rings are established and some results on exchange rings and the regularity of left SF-rings are generalized.

Beyond Earthworm: Keeping the Promise

AbstractThe founding of the Advanced National Seismic System (ANSS) vision was originally presented in U.S. Geological Survey Circular 1188 (U.S. Geological Survey [USGS], 1999), after many years of discussions and workshops, described in detail by Filson and Arabasz (2016). Much has been accomplished in the ensuing two decades. Disparate and sometimes divergent developments that had been previously explored at individual private and public universities were finally centralized with increased efficiency and coherency of effort. The stated mission of the ANSS is to “… provide accurate and timely data and information products for seismic events, including their effects on buildings and structures, employing modern monitoring methods and technologies.” In this article, an approach (xQuake) is proposed that does not interfere in any way with the mission of the National Earthquake Information Center and ANSS but instead restores much of the community focus and international collaboration that has been lost over the past two decades. xQuake uses an executable graph framework in a pipeline architecture; this framework can be seamlessly integrated into current ANSS quake monitoring systems. This new approach incorporates modern approaches to computer analytics, including multitopic Kafka exchange rings, cloud computing, a self-configuring phase associator, and machine learning. The xGraph system is free for noncommercial use, open source, hardware agnostic (Windows, Linux, Mac), with no requirement for commercial datastores.

Symmetries in idempotent factorizations

In analogy to the fact that isomorphic idempotents arise from flipped factorizations, we show that under some weak conditions, partial factorizations of the complement idempotents may also be preserved. Applying these results, we give new characterizations of exchange rings. We also show that the definition of an exchange element is left–right symmetric.

I-Reversible rings

We study rings in which [Formula: see text] nonzero idempotent implies [Formula: see text] is also an idempotent. We call such rings i-reversible. Besides studying the basic properties of i-reversible rings, we characterize i-reversible triangular matrix rings, i-reversible matrix rings over commutative rings and i-reversible exchange rings.

Weakly exchange rings whose units are sums of two idempotents

Weakly exchange rings whose units are sums of two idempotents

Some connections between frames of radical ideals and frames of z-ideals

For any semisimple f-ring A with bounded inversion, we show that the frame of radical ideals of A and the frame of z-ideals of A have isomorphic subfit coreflections. If we assume the Axiom of Choice, then the two coreflections are actually identical. If the f-ring has the property that the sum of two z-ideals is a z-ideal (as in the case of rings of continuous functions), then the epicompletion of the frame of z-ideals is shown to be a dense quotient of the epicompletion of the frame of radical ideals. Baer rings, exchange rings, and normal rings that lie in the class of semisimple f-rings with bounded inversion are shown to have characterizations in terms of frames of z-ideal which are similar to characterizations in terms of frames of radical ideals.

Additive Maps on Units of Rings

AbstractLet R be a ring. A map f: R → R is additive if f(a + b) = f(a) + f(b) for all elements a and b of R. Here, a map f: R → R is called unit-additive if f(u + v) = f(u) + f(v) for all units u and v of R. Motivated by a recent result of Xu, Pei and Yi showing that, for any field F, every unit-additive map of (F) is additive for all n ≥ z, this paper is about the question of when every unit-additivemap of a ring is additive. It is proved that every unit-additivemap of a semilocal ring R is additive if and only if either R has no homomorphic image isomorphic to or R/J(R) ≅ with 2 = 0 in R. Consequently, for any semilocal ring R, every unit-additive map of (R) is additive for all n ≥ 2. These results are further extended to rings R such that R/J(R) is a direct product of exchange rings with primitive factors Artinian. A unit-additive map f of a ring R is called unithomomorphic if f(uv) = f(u)f(v) for all units u, v of R. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.