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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]

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 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]

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 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]

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 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]

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]