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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]

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]

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]

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]

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 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]