For a finite field $mathbb{F}_q$, the bivariate skew polynomial ring $mathbb{F}_q[x,y;rho,theta]$ has been used to study codes cite{XH}. In this paper, we give some characterizations of the ring $R[x,y;rho,theta]$, where $R$ is a commutative ring. We investigate 2-D skew $(lambda_1,lambda_2)$-constacyclic codes in the ring $R[x,y;rho,theta]/langle x^l-lambda_1,y^s-lambda_2rangle_{mathit{l}}.$ A...

R e i n e r L u c k e n b a c h u n d N i k o l a u s M ü l l e r [1] Organisch-Chemisches Institut der Universität Mainz Z . Naturforsch. 34b, 464 -480 (1979); eingegangen am 10. A u g u s t / 1 9 . Dezember 1978 R ing Opening, W o o d w a r d H o f f m a n n Rules, Allylic Cations, Dimerisation, 1,5-Hexadienes React ion of cyc lopropyl halides with sodium iodide in acetone [i.e. under typical...

Let $M_R$ be a non-zero‎ ‎module and ${mathcal F}‎: ‎sigma[M_R]times sigma[M_R]‎ ‎rightarrow$ Mod-$Bbb{Z}$ a bifunctor‎. ‎The‎ ‎$mathcal{F}$-reversibility of $M$ is defined by ${mathcal‎ ‎F}(X,Y)=0 Rightarrow {mathcal F}(Y,X)=0$ for all non-zero $X,Y$‎ ‎in $sigma[M_R]$‎. ‎Hom (resp‎. ‎Rej)-reversibility of $M$ is‎ ‎characterized in different ways‎. ‎Among other things‎, ‎it is shown‎ ‎th...

$!!!!$ In this paper, the notion of fuzzy $!$ Krasner $!(m, n)$-hyperrings($!F^{(m, n)}!$-hyperrings) by using the notion of$F^m$-hyperoperations and $F^n$-operations is introduced and somerelated properties are investigated. In this regards,relationships between Krasner $F^{(m, n)}$-hyperrings and Krasner$(m, n)$-hyperrings are considered. We shall prove that everyKrasner $F^{(m, n)}$-hyperrin...

Po l lu t i on P reven t i on i s an i dea whose t i m e has come . Although the Clean Water Act, Clean Air Act, a n d e v e n S u p e r f u n d , p o i n t e d u s i n t h e r i g h t d i r e c t i o n b y i d e n t i f y i n g a r e a s o f p o l l u t i o n p r o b l e m s , t h e y p r o v e d t h a t throwing money at pol lut ion wil l not make i t go away. T r e a t m e n t i s n o t t h ...

Let $R$ be a reversible ring which is $alpha$-compatible for an endomorphism $alpha$ of $R$ and $f(X)=a_0+a_1X+cdots+a_nX^n$ be a nonzero skew polynomial in $R[X;alpha]$. It is proved that if there exists a nonzero skew polynomial $g(X)=b_0+b_1X+cdots+b_mX^m$ in $R[X;alpha]$ such that $g(X)f(X)=c$ is a constant in $R$, then $b_0a_0=c$ and there exist nonzero elements $a$ and $r$ in $R$ such tha...

Ž . We study the depth of the ring of invariants of SL F acting on the nth 2 p symmetric power of the natural two-dimensional representation for n p. These Ž . symmetric power representations are the irreducible representations of SL F 2 p over F . We prove that, when the greatest common divisor of p y 1 and n is less p than or equal to 2, the depth of the ring of invariants is 3. We also prove...

let $r$ be a reversible ring which is $alpha$-compatible for an endomorphism $alpha$ of $r$ and $f(x)=a_0+a_1x+cdots+a_nx^n$ be a nonzero skew polynomial in $r[x;alpha]$. it is proved that if there exists a nonzero skew polynomial $g(x)=b_0+b_1x+cdots+b_mx^m$ in $r[x;alpha]$ such that $g(x)f(x)=c$ is a constant in $r$, then $b_0a_0=c$ and there exist nonzero elements $a$ and $r$ in $r$ such tha...

The components of the title 1:1 co-crystal, C(14)H(10)N(4)·C(6)F(4)I(2), are connected via an N⋯I [2.959 (4) Å] halogen bond, in which the N atom is part of the relatively electron-rich pyrazine ring. The C(6)F(4)I(2) mol-ecule is almost planar [r.m.s. deviation = 0.038 Å] but there are significant twists in the pyrazine derivative, as seen in the dihedral angles [31.3 (2) and 54.6 (2)°] formed...