Exchange Ring Research Articles

Stable modules and a theorem of Camillo and Yu

In 1995, Camillo and Yu showed that an exchange ring has stable range 1 if and only if every regular element is unit-regular. An element m in a module MR is called regular if (mλ)m=m for some λ∈hom(M,R). In this paper we define stable modules and show that if M has the finite exchange property then M is stable if and only if, for every regular element m∈M, (mγ)m=m where γ:M→R is epic (and we say that m is unit-regular). Such modules are called regular-stable. It is shown that RR is regular-stable if and only if R has internal cancellation. To simplify the exposition, many arguments are formulated in an arbitrary Morita context.

USING IDEALS TO PROVIDE A UNIFIED APPROACH TO UNIQUELY CLEAN RINGS

AbstractIn this article, we introduce the notion of the uniquely $I$-clean ring and show that, if $R$ is a ring and $I$ is an ideal of $R$ then $R$ is uniquely $I$-clean if and only if ($R/ I$ is Boolean and idempotents lift uniquely modulo $I$) if and only if (for each $a\in R$ there exists a central idempotent $e\in R$ such that $e- a\in I$ and $I$ is idempotent-free). We examine when ideal extension is uniquely clean relative to an ideal. Also we obtain conditions on a ring $R$ and an ideal $I$ of $R$ under which uniquely $I$-clean rings coincide with uniquely clean rings. Further we prove that a ring $R$ is uniquely nil-clean if and only if ($N(R)$ is an ideal of $R$ and $R$ is uniquely $N(R)$-clean) if and only if $R$ is both uniquely clean and nil-clean if and only if ($R$ is an abelian exchange ring with $J(R)$ nil and every quasiregular element is uniquely clean). We also show that $R$ is a uniquely clean ring such that every prime ideal of $R$ is maximal if and only if $R$ is uniquely nil-clean ring and $N(R)= {\mathrm{Nil} }_{\ast } (R)$.

Tandem Synthesis of 9,10‐Dihydro‐9,10‐diboraanthracenes via Elusive ortho‐Lithiated Phenylboronates

AbstractA simple general protocol is reported for the synthesis of functionalised 9,10‐dihydroxy‐9,10‐dihydro‐9,10‐diboraanthracenes by treatment of the appropriate substituted (2‐halophenyl)boronates with tBuLi. The reaction apparently occurs through the formation of ortho‐lithiated phenylboronates. These elusive species possess a trivalent Lewis‐acidic boron atom in the vicinity of a strongly nucleophilic lithiated carbon atom. The results of DFT calculations indicate that Br/Li exchange is kinetically favored over nucleophilic alkylation of the boron atom, which contrasts with the behaviour of the sterically less hindered nBuLi. The results are consistent with lower yields of 9,10‐dihydro‐9,10‐diboraanthracenes obtained by using the latter reagent. Subsequent steps of the cascade reaction sequence involve trapping of the reactive lithium/boron intermediate in situ with the starting boronate, followed by a second halogen/lithium exchange and final ring closure. Isolation of 9,10‐dihydroxy‐9,10‐dihydro‐9,10‐diboraanthracenes was further aided by their complexation with 8‐hydroxyquinoline.

2-GOOD RINGS AND THEIR EXTENSIONS

P. V<TEX>$\acute{a}$</TEX>mos called a ring R 2-good if every element is the sum of two units. The ring of all <TEX>$n{\times}n$</TEX> matrices over an elementary divisor ring is 2-good. A (right) self-injective von Neumann regular ring is 2-good provided it has no 2-torsion. Some of the earlier results known to us about 2-good rings (although nobody so called at those times) were due to Ehrlich, Henriksen, Fisher, Snider, Rapharl and Badawi. We continue in this paper the study of 2-good rings by several authors. We give some examples of 2-good rings and their related properties. In particular, it is shown that if R is an exchange ring with Artinian primitive factors and 2 is a unit in R, then R is 2-good. We also investigate various kinds of extensions of 2-good rings, including the polynomial extension, Nagata extension and Dorroh extension.

ON A QUESTION OF HARTWIG AND LUH

AbstractIn 1977 Hartwig and Luh asked whether an element $a$ in a Dedekind-finite ring $R$ satisfying $aR= {a}^{2} R$ also satisfies $Ra= R{a}^{2} $. In this paper, we answer this question in the negative. We also prove that if $a$ is an element of a Dedekind-finite exchange ring $R$ and $aR= {a}^{2} R$, then $Ra= R{a}^{2} $. This gives an easier proof of Dischinger’s theorem that left strongly $\pi $-regular rings are right strongly $\pi $-regular, when it is already known that $R$ is an exchange ring.

Lifting units in clean rings

Let R be a clean ring with an ideal I such that R/I is semiperfect and K0(I) is torsion-free. We prove that, under some mild conditions, units in R/I can be lifted to units in R. This implies that the matrix ring Mn(R) over Bergmanʼs example of a non-clean exchange ring R is not clean, for every n. We also obtain some other results concerning lifting units in clean rings.

NZI rings

A ring R is called NZI if for any a \in R, l(a) is an N-ideal of R. In this paper, we first study some basic properties and basic extensions of NZI rings. Next, we study the strong regularity of NZI rings and obtain the following results: (1) Let R be a left SF-ring. Then R is a strongly regular ring if and only if R is an NZI ring; (2) If R is an NZI left MC2 ring and every simple singular left R-module is nil-injective, then R is reduced; (3) Let R be an NZI ring. Then R is a strongly regular ring if and only if R is a von Neumann regular ring; (4) Let R be an NZI ring. Then R is a clean ring if and only if R is an exchange ring.

ON PSEUDO SEMISIMPLE RINGS

A necessary and sufficient condition is obtained for a right pseudo semisimple ring to be left pseudo semisimple. It is proved that a right pseudo semisimple ring is an internal exchange ring. It is also proved that a right and left pseudo semisimple ring is an SSP ring.

Every zero adequate ring is an exchange ring

It is proved that if R is a commutative ring in which zero is an adequate element, then R is an exchange ring and every zero adequate ring is an exchange ring. There is a new description of adequate rings; this is an answer to questions formulated by Larsen, Lewis, and Shores.

On Rings with the Goodearl–Menal Condition

A ring R is said to satisfy the Goodearl–Menal condition if for any x, y ∈ R, there exists a unit u of R such that both x − u and y − u −1 are units of R. It is proved that if R is a semilocal ring or an exchange ring with primitive factors Artinian, then R satisfies the Goodearl–Menal condition if and only if no homomorphic image of R is isomorphic to either ℤ2 or ℤ3 or 𝕄2(ℤ2). These results correct two existing results. Some consequences are discussed.

Quasiunit regularity and QB-rings

Some relations for quasiunit regular rings and QB-rings, as well as for pseudounit regular rings and QB ∞-rings, are obtained. In the first part of the paper, we prove that (an exchange ring R is a QB-ring) ⟺ (whenever x ∈ R is regular, there exists a quasiunit regular element w ∈ R such that x = xyx = xyw for some y ∈ R) ⟺ (whenever aR + bR = dR in R; there exists a quasiunit regular element w ∈ R such that a + bz = dw for some z ∈ R). Similarly, we also give necessary and sufficient conditions for QB ∞-rings in the second part of the paper.

Extensions and s-comparability of exchange rings

Let S be a ring extension of R. In this note, for any positive integer s we study s-comparability related to ring extensions. We show that if S is an excellent extension of R, R and S are exchange rings, and R has the n-unperforation property. R satisfies s-comparability if and we only if so does S, and we prove that for a 2-sided ideal J of S, and an exchange subring R of the exchange ring S, which contains J as a direct summand, then R satisfies s-comparability if and only if so does R/J.

Lifting Defects for Nonstable K0-theory of Exchange Rings and C*-algebras

The assignment (nonstable K0-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: By using categorical tools (larders, lifters, CLL) from a recent book from the author with P. Gillibert, we deduce that there exists a unital exchange ring of cardinality $\aleph_3$ (resp., an $\aleph_3$ -separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V( R ) is the positive cone of a dimension group but it is not isomorphic to V( B ) for any ring B which is either a C*-algebra of real rank 0 or a regular ring.

A novel bartering exchange ring based incentive mechanism for peer-to-peer systems

Peer-to-Peer (P2P) networking is an alternative to the cloud computing for relatively more informal trade. One of the major obstacles to its development is the free riding problem, which significantly degrades the scalability, fault tolerance and content availability of the systems. Bartering exchange ring based incentive mechanism is one of the most common solutions to this problem. It organizes the users with asymmetric interests in the bartering exchange rings, enforcing the users to contribute while consuming. However the existing bartering exchange ring formation approaches have inefficient and static limitations. This paper proposes a novel cluster based incentive mechanism (CBIM) that enables dynamic ring formation by modifying the Query Protocol of underlying P2P systems. It also uses a reputation system to alleviate malicious behaviors. The users identify free riders by fully utilizing their local transaction information. The identified free riders are blacklisted and thus isolated. The simulation results indicate that by applying the CBIM, the request success rate can be noticeably increased since the rational nodes are forced to become more cooperative and the free riding behaviors can be identified to a certain extent.

On Uniquely Clean Rings

A ring R is uniquely (nil) clean in case for any a ∈ R there exists a unique idempotent e ∈ R such that a − e ∈ R is invertible (nilpotent). We prove in this article that a ring R is uniquely clean if and only if R is an exchange ring with all idempotents central, and that for all maximal ideals M of R, R/M ≅ ℤ2. It is shown that every uniquely clean ring of which every prime ideal of R is maximal is uniquely nil clean. Further, we prove that a ring R is uniquely nil clean if and only if R/J(R) is Boolean and R is a π-regular ring with all idempotents central.

Strong lifting splits

The concept of an enabling ideal is introduced so that an ideal I is strongly lifting if and only if it is lifting and enabling. These ideals are studied and their properties are described. It is shown that a left duo ring is an exchange ring if and only if every ideal is enabling, that Zhou’s δ -ideal is always enabling, and that the right singular ideal is enabling if and only if it is contained in the Jacobson radical. The notion of a weakly enabling left ideal is defined, and it is shown that a ring is an exchange ring if and only if every left ideal is weakly enabling. Two related conditions, interesting in themselves, are investigated: the first gives a new characterization of δ -small left ideals, and the second characterizes weakly enabling left ideals. As an application (which motivated the paper), let M be an I -semiregular left module where I is an enabling ideal. It is shown that m ∈ M is I -semiregular if and only if m − q ∈ I M for some regular element q of M and, as a consequence, that if M is countably generated and I M is δ -small in M , then M ≅ ⊕ i = 1 ∞ R e i where e i 2 = e i ∈ R for each i .

PIERCE STALKS OF EXCHANGE RINGS

We prove, in this article, that a ring R is a stable exchange ring if and only if so are all its Pierce stalks. If every Pierce stalks of R is artinian, then <TEX>$1_R$</TEX> = u + <TEX>$\upsilon$</TEX> with u, <TEX>$\upsilon$</TEX> <TEX>$\in$</TEX> U(R) if and only if for any a <TEX>$\in$</TEX> R, there exist u, <TEX>$\upsilon$</TEX> <TEX>$\in$</TEX> U(R) such that a = u + <TEX>$\upsilon$</TEX>. Furthermore, there exists u <TEX>$\in$</TEX> U(R) such that <TEX>$1_R\;{\pm}\;u\;\in\;U(R)$</TEX> if and only if for any a <TEX>$\in$</TEX> R, there exists u <TEX>$\in$</TEX> U(R) such that <TEX>$a\;{\pm}\;u\;\in\;U(R)$</TEX>. We will give analogues to normal exchange rings. The root properties of such exchange rings are also obtained.

Quasi-Normal Rings

A ring R is defined to be quasi-normal if ae = 0 implies eaRe = 0 for a ∈ N(R) and e ∈ E(R), where E(R) and N(R) stand, respectively, for the set of idempotents and the set of nilpotents of R. It is proved that R is quasi-normal if and only if eR(1 − e)Re = 0 for each e ∈ E(R) if and only if T n (R, R) is quasi-normal for any positive integer n. And it follows that for a quasi-normal ring R, R is Π-regular if and only if N(R) is an ideal of R and R/N(R) is regular. Also, using quasi-normal ring, we proved the following: (1) R is an abelian ring if and only if R is a quasi-normal left idempotent reflexive ring; (2) R is a strongly regular ring if and only if R is a von Neumann regular quasi-normal ring; (3) Let R be a quasi-normal ring. Then R is a clean ring if and only if R is an exchange ring; (4) Let R be a quasi-normal Π-regular ring. Then R is a (S,2)-ring if and only if ℤ/2ℤ is not a homomorphic image of R.

Induced liftings, exchange rings and semi-perfect algebras

Several important classes of rings can be characterized in terms of liftings of idempotents with respect to various ideals: classical examples are semi-perfect rings, semi-regular rings and exchange rings. We begin with a study of some extensions of the concept of idempotent lifting and prove the generalizations of some classical lifting theorems. Then we describe the method of induced liftings, which allows us to transfer liftings from a ring to its subrings. Using this method we are able to show that under certain assumptions a subring of an exchange ring is also an exchange ring, and to prove that a finite algebra over a commutative local ring is semi-perfect, provided it can be suitably represented in an exchange ring.

On Exchange Hermitian Rings

In this article, we investigate new necessary and sufficient conditions on an exchange ring under which every regular matrix admits a diagonal reduction. We prove that an exchange ring R is an hermitian ring if and only if for any n ≥ 2 and any regular x ∈ Rn, there exists u ∈ CLn(R) such that x = xux; if and only if for any n ≥ 2 and any regular x ∈ Rn, there exists u ∈ CLn(R) such that xu ∈ R is an idempotent. Furthermore, we characterize such exchange rings by means of reflexive inverses and n-pseudo-similarity.