we introduce center-like subsets z*(r,f), z**(r,f) and z1(r,f), where r is a ring and f is a map from r to r. for f a derivation or a non-identity epimorphism and r a suitably-chosen prime or semiprime ring, we prove that these sets coincide with the center of r.

for an arbitrary ring $r$, the zero-divisor graph of $r$, denoted by $gamma (r)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $r$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. it is well-known that for any commutative ring $r$, $gamma (r) cong gamma (t(r))$ where $t(r)$ is the (total) quotient ring of $r$. in this...

let r be a prime ring with extended centroid c, h a generalized derivation of r and n ⩾ 1 a xed integer. in this paper we study the situations: (1) if (h(xy))n = (h(x))n(h(y))n for all x; y 2 r; (2) obtain some related result in case r is a noncommutative banach algebra and h is continuous or spectrally bounded.

a ring $r$ is a strongly clean ring if every element in $r$ is the sum of an idempotent and a unit that commutate. we construct some classes of strongly clean rings which have stable range one. it is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.

in this paper we define a new type of rings ”almost powerhermitian rings” (a generalization of almost hermitian rings) and establish several sufficient conditions over a ring r such that, every regular matrix admits a diagonal power-reduction.

We define a closure operation for rings of mixed characteristic and verify that the closure is a ring. We then show that this closure produces a ring with good properties with respect to its Fontaine ring and give an example to show that rings that are not closed in this sense do not satisfy these properties.

Rh(II)-catalyzed diversified ring expansions controlled by single-electron-transfer (SET) have been disclosed in this communication, producing a series of indole-fused azetidines and 1H-carbazoles or related derivatives in moderate to good yields via Rh2(III,II) nitrene radical intermediates. The direction of ring expansion branches according to different ring sizes of methylenecycloalkanes.

derivatives of thiazolidinone ring systems are known to act as anti-hiv infections , analgesic, anti-bacterial, anti-convulsant, anti parasitic, potential anti-inflammatory and herbicidal agents. due biological activities of thiazolidinones ring, several methods for their synthesis have been illustrated in literature. imidazolidine-2-thiones were synthesized by the oxidative cyclization of 1-be...

The ring expansion of small-ring compounds provides a powerful method for the construction of various cyclic compounds. Herein, nitrene equivalent mediated metal-free ring expansions of alkylidenecyclopropanes (ACPs) and an alkylidenecyclobutane (ACB) were described. In this synthesis, a series of aryl-substituted cyclobutylidene and cyclopentylidene hydrazine derivatives were obtained under mi...