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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 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]

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]

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]

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]