for a ring endomorphism $alpha$ and an $alpha$-derivation $delta$, we introduce a concept, so called skew $pi$-armendariz ring, that is a generalization of both $pi$-armendariz rings, and $(alpha,delta)$-compatible skew armendariz rings. we first observe the basic properties of skew $pi$-armendariz rings, and extend the class of skew $pi$-armendariz rings through various ring extensions. we nex...

an efficient one-pot three-component procedure for the synthesis of new chiral spiro oxindolopyrrolidines/pyrrolizidines with highly regio- and diastereo-enantio, selective from 1,3-dipolar cycloaddition of azomethine ylides and chiral menthol-drived trans-cinnamic are described. the mechanism of the reaction is discussed on basis of the assignment of the absolute configuration of one of the cy...

in this paper, we introduce the new notion of n-f-clean rings as a generalization of f-clean rings. next, we investigate some properties of such rings. we prove that mn(r) is n-f-clean for any n-f-clean ring r. we also, get a condition under which the denitions of n-cleanness and n-f-cleanness are equivalent.

a topoframe, denoted by $l_{ tau}$,  is a pair $(l, tau)$ consisting of a frame $l$ and a subframe $ tau $ all of whose elements are complementary elements in$l$. in this paper, we define and study the notions of a$tau $-real-continuous function on a frame $l$ and the set of realcontinuous functions $mathcal{r}l_tau $ as an $f$-ring.we show that $mathcal{r}l_{ tau}$is actually a generalization ...

for a fixed positive integer , we say a ring with identity is n-generalized right principally quasi-baer, if for any principal right ideal of , the right annihilator of is generated by an idempotent. this class of rings includes the right principally quasi-baer rings and hence all prime rings. a certain n-generalized principally quasi-baer subring of the matrix ring are studied, and connections...

chiral pyrrolidines and pyrrolizidines with spirooxindole ring systems are the central skeletons for numerous alkaloids and pharmacologically important compounds. gelesmine, pseudotabersonine, formosanine, isoformosanine, morroniside and mitraphylline are some of the alkaloids containing spirooxindole ring systems. derivatives of spirooxindole find very wide biological applications as anti micr...

Chiral pyrrolidines and pyrrolizidines with spirooxindole ring systems are the central skeletons for numerous alkaloids and pharmacologically important compounds. Gelesmine, pseudotabersonine, formosanine, isoformosanine, morroniside and mitraphylline are some of the alkaloids containing spirooxindole ring systems. Derivatives of spirooxindole find very wide biological applications as anti micr...

An efficient one-pot three-component procedure for the synthesis of new chiral spiro oxindolopyrrolidines/pyrrolizidines with highly regio- and diastereo-enantio, selective from 1,3-dipolar cycloaddition of azomethine ylides and chiral menthol-drived trans-cinnamic are described. The mechanism of the reaction is discussed on basis of the assignment of the absolute configuration of one of the cy...