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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 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]

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]