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A ring element a ∈ R is said to be of right stable range one if, for any t ∈ R , a R + t R = R implies that a + t b is a unit in R for some b ∈ R. Similarly, a ∈ R is said to be of left stable range one if R a + R t = R implies that a + b ′ t is a unit in R for some b ′ ∈ R. In the last two decades, it has often been speculated that these two notions are actually the same for any a ∈ R. In §3 of this paper, we will prove that this is indeed the case. The key to the proof of this new symmetry result is a certain "Super Jacobson's Lemma", which generalizes Jacobson's classical lemma stating that, for any a , b ∈ R , 1 − a b is a unit in R iff so is 1 − b a. Our proof for the symmetry result above has led to a new generalization of a classical determinantal identity of Sylvester, which will be published separately in [45]. In §§4-5, a detailed study is offered for stable range one ring elements that are unit-regular or nilpotent, while §6 examines the behavior of stable range one elements via their classical Peirce decompositions. The paper ends with a more concrete §7 on integral matrices of stable range one, followed by a final §8 with a few open questions. [ABSTRACT FROM AUTHOR]

For any three n × n matrices A,B,X over a commutative ring S, we prove that det(A + B − AXB) = det(A + B − BXA) ∈ S. This apparently new formula may be regarded as a “ternary generalization” of Sylvester’s classical determinantal formula det(In − AB) = det(In − BA) for any pair of n × n matrices A,B over S. By suitably specializing our ternary formula above, we have also obtained several other binary determinantal and tracial identities that are hitherto unknown. [ABSTRACT FROM AUTHOR]

We prove that a ring R is an $n \times n$ matrix ring (that is, $R \cong \mathbb {M}_n(S)$ for some ring S) if and only if there exists a (von Neumann) regular element x in R such that $l_R(x) = R{x^{n-1}}$. As applications, we prove some new results, strengthen some known results and provide easier proofs of other results. For instance, we prove that if a ring R has elements x and y such that $x^n = 0$ , $Rx+Ry = R$ and $Ry \cap l_{R}(x^{n-1}) = 0$ , then R is an $n \times n$ matrix ring. This improves a result of Fuchs ['A characterisation result for matrix rings', Bull. Aust. Math. Soc. 43 (1991), 265–267] where it is proved assuming further that the element y is nilpotent of index two and $x+y$ is a unit. For an ideal I of a ring R , we prove that the ring $(\begin {smallmatrix} R & I \\ R & R \end {smallmatrix})$ is a $2 \times 2$ matrix ring if and only if $R/I$ is so. [ABSTRACT FROM AUTHOR]

We find an expression for the Drazin inverse of a strongly π -regular element a in the form a i x j when a n = a n + k x for some nonnegative integer n and positive integer k. This extends the result by Azumaya, which is the case when n = 1 and k = 1 , and a result by Drazin, which is the case when n is arbitrary and k = 1. We give new proofs of several results in the literature. For instance, we give an easy proof of the result that for two commuting Drazin invertible elements a , b of R , then a + b is Drazin invertible if and only if so is 1 + a D b , where a D is the Drazin inverse of a. Our proof is akin to the case when a and b are invertible. [ABSTRACT FROM AUTHOR]

We investigate connections between association chains of idempotents and bounded generation of SL 2 by elementary matrices. This enables us to identify many classes of matrix rings that have universal bounds on the length of association chains. In particular, many matrix rings have the property that every regular element is special clean. We also improve the usual criterion for checking perspectivity via association chains. [ABSTRACT FROM AUTHOR]

A clean decomposition a = e + u in a ring R (with idempotent e and unit u) is said to be special if a R ∩ e R = 0. We show that this is a left-right symmetric condition. Special clean elements (with such decompositions) exist in abundance, and are generally quite accessible to computations. Besides being both clean and unit-regular, they have many remarkable properties with respect to element-wise operations in rings. Several characterizations of special clean elements are obtained in terms of exchange equations, Bott–Duffin invertibility, and unit-regular factorizations. Such characterizations lead to some interesting constructions of families of special clean elements. Decompositions that are both special clean and strongly clean are precisely spectral decompositions of the group invertible elements. The paper also introduces a natural involution structure on the set of special clean decompositions, and describes the fixed point set of this involution. [ABSTRACT FROM AUTHOR]

We study rings in which ab nonzero idempotent implies ba 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. [ABSTRACT FROM AUTHOR]

It is well-known that idempotents lift modulo any nil one-sided ideal. While this is not true for periodic elements, it does hold true in many special cases. We investigate the connections between these special cases, as well as limitations. We also answer three questions from the literature. For example, we construct a nilpotent ideal where torsion-units lift, but periodic elements do not lift, modulo that ideal. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The equation XA-BX = I has been well studied in ring theory, operator theory, linear algebra, and other branches of mathematics. In this paper, we show that, in the case where B2 = B, the study of XA - BX = I in a noncommutative ring R leads to several new ways to view and to work with the exchange (or "suitable") elements in R in the sense of Nicholson. For any exchange element A ∊ R, we show that the set of idempotents E ∊ R such that E ∊ RA and I - E ∊ R(I - A) is naturally parametrized by the roots of a certain left-right symmetric "exchange polynomial" associated with A. From the new viewpoints on exchange elements developed in this paper, the classes of clean and strongly clean elements in rings can also be better understood. [ABSTRACT FROM AUTHOR]

In this paper, we study exchange rings and clean rings with (or otherwise). Analogues of a theorem of Camillo and Yu characterizing clean and strongly clean rings with are obtained for such rings (as well as for exchange rings) using the viewpoint of exchange equations introduced in a recent paper of the authors. We also study a new class of rings including von Neumann regular rings in which square roots of one (instead of idempotents) can be lifted modulo left ideals, and conjecture that such rings are exchange rings. This conjecture holds for commutative rings, and would hold for all rings if it holds for semiprimitive rings of characteristic . [ABSTRACT FROM AUTHOR]

We prove that if no field of order less than n + 2 is a homomorphic image of a right self-injective ring R, then for any element a ∈ R and central units u1, u2, ..., un ∈ U(R) there exists a unit u ∈ U(R) such that a + uiu ∈ U(R) for each i ≥ 1. [ABSTRACT FROM AUTHOR]

In 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. [ABSTRACT FROM PUBLISHER]

An element of the form [a, b, c] = abc − cba in a ring is called a generalized commutator. In this article, we show that, in a matrix ring 𝕄 n (S) (n ≥ 2) over any ring S (with identity), every matrix is a sum of a commutator and a generalized commutator. If S is an elementary divisor ring (in the sense that every square matrix over S is equivalent to a diagonal matrix), then every matrix in 𝕄 n (S) (n ≥ 2) is, in fact, a generalized commutator. Using suitable generic techniques, we show, however, that this conclusion is not true in general for various rings S (e.g. polynomial rings and power series rings). Indeed, for any n ≥ 2, there exists an n × n traceless matrix over some commutative ring S that is not a generalized commutator (respectively, is a generalized commutator but not a commutator) in 𝕄 n (S). This gives, in particular, a strong negative solution to the problem whether n × n traceless matrices are necessarily commutators over a commutative base ring, for any n ≥ 2. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we establish a determinantal formula for matrix commutators over a commutative ring, using (among other invariants) the quantum traces of and . Special forms of this determinantal formula include a “trace version”, and a “supertrace version”. Some applications of these formulas are given to the study of value sets of binary quadratic forms, the factorization of integral matrices, and the solution of certain simultaneous diophantine equations over commutative rings. [Copyright &y& Elsevier]

Abstract: A ring S is said to have square stable range one (written ) if implies that is a unit for some . In the commutative case, this extends the class of rings of stable range 1, and allows many new examples such as rings of real-valued continuous functions, and real holomorphy rings. On the other hand, sometimes forces S to have stable range 1. For instance, this is the case for exchange rings S, for which is characterized by being reduced (or abelian, or right quasi-duo). We also characterize rings S whose (von Neumann) regular elements are strongly regular, by using an element-wise notion of square stable range one. Extending a result of Estes and Ohm, we show that a possibly noncommutative infinite domain with stable range one or square stable range one must have a non-artinian group of units. [Copyright &y& Elsevier]

In a matrix ring R = 필2(S) where S is a commutative ring, we study equations of the form XY - YX = U ∈ GL2(S), focusing on matrices in R that can appear as X or as XY in such equations. These are the completable and the reflectable matrices in R. For matrices A ∈ R with a zero row or with a constant diagonal, explicit and "computer-checkable" criteria are found for A to be completable or reflectable. A formula for det (XY - YX) discovered recently with Shomron connects this study to diophantine questions about the representation of units of the ground ring S by quadratic forms of the type px2 + qy2. [ABSTRACT FROM AUTHOR]

A ring R is said to be von Newmann local (VNL) if for any a ∈ R, either a or 1 -a is (von Neumann) regular. The class of VNL rings lies properly between exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize and classify arbitrary VNL rings without an infinite set of orthogonal idempotents; and also the VNL rings having a primitive idempotent e such that eRe is not a division ring. We prove that a semiperfect ring R is VNL if and only if for any right uni-modular row (a1, a2) ∈ R2, one of the ai's is regular in R. Formal triangular matrix rings that are VNL are also characterized. As a corollary, it is shown that an upper triangular matrix ring Tn(R) is VNL if and only if n = 2 or 3 and R is a division ring. [ABSTRACT FROM AUTHOR]

Let R be a ring and M(R) the set consisting of zero and primitive idempotents of R. We study the rings R for which M(R) is multiplicative. It is proved that if R has a complete finite set of primitive orthogonal idempotents, then R is a finite direct product of connected rings precisely when M(R) is multiplicative. We prove that if R is a (von Neumann) regular ring with M(R) multiplicative, then every primitive idempotent in R is central. It is also shown that this does not happen even in semihereditary and semiregular rings. Let R be an arbitrary ring with M(R) multiplicative and e ∈ R be a primitive idempotent, then for every unit u ∈ R, it is proved that eue is a unit in eRe. We also prove that if M(R) is multiplicative, then two primitive idempotents e and f in R are conjugates, i.e., f = ueu-1 for some u ∈ U(R), if and only if ef ≠ 0. [ABSTRACT FROM AUTHOR]

It is classically known that idempotents lift modulo nil one-sided ideals. So it is natural to ask if the same is true for potent elements. Although we answer this question in negative, we prove that the answer is positive in several special cases. For instance, the answer is positive in rings with finite characteristic. Let I be a nil one sided ideal of R and x ∈ R is such that x n + 1 − x ∈ I for some positive integer n. We prove that if n is a unit in R , then there exists an element p ∈ R with p n + 1 = p such that x − p ∈ I. By taking n = 1 , the result that idempotents lift modulo nil one sided ideals is retrieved. For a nil ideal I of an abelian ring R , we prove that potent elements lift modulo I precisely when torsion units or periodic elements lift modulo I. It follows that torsion units or periodic elements may also not lift modulo nil ideals. We prove that torsion units, potent elements and periodic elements lift modulo every nil ideal of a π -regular ring. [ABSTRACT FROM AUTHOR]

A classical result of Zelinsky states that every linear transformation on a vector space V, except when V is one-dimensional over ℤ2, is a sum of two invertible linear transformations. We extend this result to any right self-injective ring R by proving that every element of R is a sum of two units if no factor ring of R is isomorphic to ℤ2. [ABSTRACT FROM AUTHOR]

Abstract: An element in a ring R is said to be clean (respectively unit-regular) if it is the sum (respectively product) of an idempotent element and an invertible element. If all elements in R are unit-regular, it is known that all elements in R are clean. In this note, we show that a single unit-regular element in a ring need not be clean. More generally, a criterion is given for a matrix to be clean in a matrix ring over any commutative ring K. For , this criterion shows, for instance, that the unit-regular matrix is not clean. Also, this turns out to be the “smallest” such example. [Copyright &y& Elsevier]

It is proved that a ring R is unit regular if and only if every element a of R can be written as e + u such that aR ∩ eR = 0, where e is an idempotent and u a unit in R. [ABSTRACT FROM AUTHOR]

Examines the endomorphism rings of the Harada modules. Proof of the Kasch's Theorem using the property of Harada modules; Direct summands of the Harada module; Notations and terminology on the full linear ring.

In this paper, we study the modules whose any two isomorphic summands have a common complement. We call such modules perspective. It is proved that perspectivity is an ER-property, in the sense that it depends only on the endomorphism ring of the module. This property in rings turns out to be left-right symmetric, that is, RR is perspective if and only if RR is perspective for any ring R and we call such ring a perspective ring. The class of perspective rings lies properly between the class of rings with stable range one and the class of rings with the internal cancellation property. We give several characterizations of perspective rings and also study perspectivity in related rings. It is proved that R has stable range one if and only if M2(R) is a perspective ring. Another characterization of rings with stable range one, in terms of completions of unimodular rows, is also given. We prove that every regular element in a perspective ring is clean. [ABSTRACT FROM AUTHOR]

In this paper, we study rings that are additively generated by units. We prove that if the identity in a ring with stable range one is a sum of two units, then every (von Neumann) regular element is a sum of two units. It follows that every element in a unit-regular ring is a sum of two units if the identity is a sum of two units. Also, if the identity of a strongly π-regular ring is a sum of two units, then every element is a sum of three units. [ABSTRACT FROM AUTHOR]

Abstract: We prove that the transpose of every invertible square matrix over a ring R is invertible if and only if is commutative. Many other characterizations are obtained for such rings R in terms of (the group of units of R), including, for instance, , and (for all ). We also consider a natural weakening of these conditions, namely, , and show that, for von Neumann regular rings, this is a (necessary and) sufficient condition for the commutativity of R. [Copyright &y& Elsevier]

Abstract: Many authors have investigated the behavior of strong cleanness under certain ring extensions. In this note, we investigate the classical problem of lifting idempotents, in order to consolidate and extend these results. Our main result is that if is a ring which is complete with respect to an ideal and if is an element of whose image in is strongly -regular, then is strongly clean in . This generalizes Theorem 2.1 of Chen and Zhou (2007) [9]. [Copyright &y& Elsevier]