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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 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 study matrices over arbitrary rings that are integral over the base ring. For any ring R , let f , g , h ∈ R [ x 1 , x 2 , ... , x n ] with h = f g. We prove that for any pairwise commuting elements a 1 , a 2 , ... , a n ∈ R if g (a 1 , a 2 , ... , a n) = 0 , then h (a 1 , a 2 , ... , a n) = 0. As a corollary, it follows that for f ∈ n (S) [ x 1 , x 2 , ... , x n ] , S commutative ring, if A 1 , A 2 , ... , A n ∈ n (S) are pairwise commuting matrices such that f (A 1 , A 2 , ... , A n) = 0 , then g (A 1 , A 2 , ... , A n) = 0 where g = det (f) I. This result, which is a generalization of the Cayley–Hamilton Theorem, was proved by Phillips in 1919. For a positive integer n > 1 , we prove that if every matrix in n (R) satisfies a monic polynomial of degree n over R , then R is commutative. On the other hand, every diagonal matrix in n (R) , n > 1 , satisfies a monic polynomial of degree n over R precisely when R is a left duo ring. We prove that if every diagonal matrix in n (R) , n > 1 , is R -integral, then R is Dedekind finite. [ABSTRACT FROM AUTHOR]

We study modules in which perspectivity of summands is transitive. Generalizing a 1977 result of Handelman and a 2014 result of Garg, Grover, and Khurana, we prove that for any ring R, perspectivity is transitive in M 2 (R) if and only if R has stable range one. Also generalizing a 2019 result of Amini, Amini, and Momtahan we prove that a quasi-continuous module in which perspectivity is transitive is perspective. [ABSTRACT FROM AUTHOR]

We investigate connections between von Neumann regularity of endomorphisms and perspectivity of direct summands in modules. This leads to a new classification of those rings whose regular elements are strongly regular, which turn out to be exactly the rings R whose idempotents are central modulo the Jacobson radical J(R). An important component of our work is an investigation of the left and right associate relations on idempotents, as well as chains of these relations. As applications we give new characterizations of strongly regular elements and of idempotents that are central modulo the Jacobson radical. We also introduce a new class of regular elements that we call pc-regular elements, related to perspectivity in complement summands. These pc-regular elements are exactly the special clean elements. Generalizing the well-known fact that unit-regular rings are special clean, we then show that the unit-regular elements of any regular ring satisfying general comparability are special clean. Consequently, unit-regular endomorphisms of quasi-continuous modules are special clean, answering, in the positive, a conjecture of Lam. [ABSTRACT FROM AUTHOR]

We will give some glimpses into noncommutative ring theory by discussing some typical results. One of the aim is to demonstrate some basic ways in which noncommutative ring theory is different from its commutative version. [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]

In 1942, McCoy proved that if a polynomial f(x) over a commutative ring R has a nonzero annihilator in R [ x ] , then it has a nonzero annihilator in R also. Later in 1957 McCoy proved that for f (x) ∈ R [ x ] , R an arbitrary ring, if f (x) R [ x ] is annihilated on the right by some nonzero polynomial in R [ x ] , then f (x) R [ x ] is also annihilated on the right by some nonzero element of R. Generalizing these results of McCoy we prove that a polynomial f(x) over an arbitrary ring R has a nonzero right annihilator in R if and only if there exists a nonzero polynomial g(x) in R [ x ] such that f (x) cg (x) = 0 for every c in the multiplicative monoid generated by the coefficients of f(x). We also show that taking c to be identity and coefficients of f(x) does not suffice. We also give a new proof of Nielsen's result that every reversible ring is right McCoy. We prove that an exchange right linearly McCoy ring is reduced modulo the Jacobson radical. We also prove that R = ( S M 0 T ) is a McCoy ring, where S and T are local rings with nonzero Jacobson radicals in which product of two non-units is zero and S M T is a bimodule such that J (S) M = 0 and MJ(T) = 0. This leads to a large class of exchange non-abelian McCoy rings. [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]

Let a be a regular element of a ring R. If either K:=r( a) has the exchange property or every power of a is regular, then we prove that for every positive integer n there exist decompositions where $Y_{n} \subseteq a^{n}R$ and E ≅ R/ a R. As applications we get easier proofs of the results that a strongly π-regular ring has stable range one and also that a strongly π-regular element whose every power is regular is unit-regular. [ABSTRACT FROM AUTHOR]

For any element a in an exchange ring R, we show that there is an idempotent $\,e\in aR\cap R\,a\,$ such that $\,1-e\in (1-a)\,R\cap R\,(1-a)$. A closely related result is that a ring R is an exchange ring if and only if, for every a∈ R, there exists an idempotent e∈ R a such that 1− e∈(1− a) R. The Main Theorem of this paper is a general two-sided statement on exchange elements in arbitrary rings which subsumes both of these results. Finally, applications of these results are given to the study of the endomorphism rings of exchange modules. [ABSTRACT FROM AUTHOR]

It is well known that every uniquely clean ring is strongly clean. In this article, we investigate the question of when this result holds element-wise. We first construct an example showing that uniquely clean elements need not be strongly clean. However, in case every corner ring is clean the uniquely clean elements are strongly clean. Further, we classify the set of uniquely clean elements for various classes of rings, including semiperfect rings, unit-regular rings, and endomorphism rings of continuous modules. [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]

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]

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]

We show that in a ring of stable range 1, any (von Neumann) regular element is clean. Our main results also imply that any unit-regular ring has idempotent stable range 1 (and is therefore clean), and that a semilocal ring has idempotent stable range 1 if and only if it is semiperfect. [ABSTRACT FROM AUTHOR]

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 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]

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. 'Classroom' is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science. [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.