‎let $r$ be a noncommutative prime ring of‎ ‎characteristic different from $2$‎, ‎$u$ the utumi quotient ring of $r$‎, ‎$c$ $(=z(u))$ the extended centroid‎ ‎of $r$‎. ‎let $0neq ain r$ and $f(x_1,ldots,x_n)$ a multilinear‎ ‎polynomial over $c$ which is noncentral valued on $r$‎. ‎suppose‎ ‎that $g$ and $h$ are two nonzero generalized derivations of $r$‎ ‎such that $a(h(f(x))f(x)-f(x)g(f(x)))in ...

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, F and G non-zero generalized derivations of R. If the composition (FG) acts as a Lie derivation on R, then (FG) is a derivation of R and one of the following holds: 1. there exist α ∈ C and a ∈ U such that F (x) = [a, x] and G(x) = αx, for all x ∈ R; 2. G is an usual derivatio...

‎Let $R$ be a noncommutative prime ring of‎ ‎characteristic different from $2$‎, ‎$U$ the Utumi quotient ring of $R$‎, ‎$C$ $(=Z(U))$ the extended centroid‎ ‎of $R$‎. ‎Let $0neq ain R$ and $f(x_1,ldots,x_n)$ a multilinear‎ ‎polynomial over $C$ which is noncentral valued on $R$‎. ‎Suppose‎ ‎that $G$ and $H$ are two nonzero generalized derivations of $R$‎ ‎such that $a(H(f(x))f(x)-f(x)G(f(x)))in ...

A Tychonoff space X such that the maximum ring of quotients of C(X) is uniformly complete is called a uniform quotients space. It is shown that this condition is equivalent to the Dedekind– MacNeille completion of C(X) being a ring of quotients of C(X), in the sense of Utumi. A compact metric space is a uniform quotients space precisely when it has a dense set of isolated points. Extremally dis...

In this paper, we define topological hyperrings and study their basic concepts which supported by illustrative examples. We show some differences between topological rings and topological hyperrings. Also, by the fundamental relation $\Gamma^{*}$, we indicate the role of complete parts (saturated subsets) and complete hyperrings in topological hyperrings and specially we show that if every (clo...

1. Let QiX) denote the maximal ring of quotients (in the sense of Johnson [4] and Utumi [5]) of the ring C(X) of continuous realvalued functions on the completely regular Hausdorff space X, This ring has been studied by Fine, Gillman, and Lambek [ l] and realized by them as the direct limit of the subrings C(V), Va, dense open subset of X (i.e., the union of these C(F)'s, modulo the obvious equ...

the concepts of fuzzy semi-ideals of r with respect to h≤r and generalized fuzzy quotient rings are introduced. some properties of fuzzy semiideals are discussed. finally, several isomorphism theorems for generalized fuzzy quotient rings are established.

In this paper, we present different results related to bezoutian and residue theory. We consider, in particular, the problem of computing the structure of the quotient ring by an affine complete intersection, and an algorithm to obtain it, as conjectured in [9]. We analyze it in detail and prove the validity of the conjecture, for a modification of the initial method. Direct applications of the...