‎let $r$ be a ring with involution $*$‎. ‎an additive mapping $t:rto r$ is called a left(respectively right) centralizer if $t(xy)=t(x)y$ (respectively $t(xy)=xt(y)$) for all $x,yin r$‎. ‎the purpose of this paper is to examine the commutativity of prime rings with involution satisfying certain identities involving left centralizers.

6-methyl-2-phenyl-7h-oxazolo [3,2-b] [1,2,4] triazine (1) undersent ting transformation on treatment with ammonia and primary amines to afford the corresponding imidazol [1,2-b][1,2,4] triazine (2). treatment of (1) with hydrazine hydrate gave the corresponding 3-aryl-4h-1,2,4- triazino [4,3-b][1,2,4] triazin-8 (1h)-one (3).

in this paper, we define and study the notion of the real-valued functions on a frame $l$. we show that $f(l) $, consisting of all frame homomorphisms from the power set of $mathbb{r}$ to a frame $ l$, is an $f$-ring, as a generalization of all functions from a set $x$ into $mathbb r$. also, we show that $f(l) $ is isomorphic to a sub-$f$-ring of $mathcal{r}(l)$, the ring of real-valued continu...

abstract determining saturated hydraulic conductivity (ks) is one of the important soil characteristics that its estimation is important in soil and water studies. there are several methods for estimating ks in unsaturated conditions. these methods include the inversed-auger-hole method, the double-ring infiltrometer and the single-ring pressure infiltrometer. the measurement of ks by these met...