Let $F$ be a non-negatively graded free module over polynomial ring $\mathbb{K}[x_1,\dots,x_n]$ generated by $m$ basis elements. $M$ submodule of elements in with degrees bounded $D$ and dim $F/M$=$r$. We prove that if is graded, the degree reduced Gr\"{o}bner for any term order $2\left[1/2((Dm)^{n-r}m+D) \right]^{2^{r-1}}$. If not bound $2\left[1/2((Dm)^{(n-r)^2}m+D) \right]^{2^{r}}$. This gen...

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...

An element is known a strongly nil* clean if a=e1 - e1e2 + n , where e1,e2 are idempotents and nilpotent, that commute with one another. ideal I of ring R called each element. We investigate some its fundamental features, as well relationship to the nil ideal.

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$L$. $f$-ring $mathcal{R}(L_{ tau})$ is equal to the set $${fin Frm(mathcal{P}(mathbb R), L): f(mathfrak{O}(mathbb R))subseteq tau} .$$ In this paper, for every complemented element $ain L$ with $a, a'...

In the title mol-ecule, C(18)H(16)F(3)N(5)O(2), the dihedral angle between the pyridine ring and the fused benzene ring is 4.50 (10)°. The triazole ring makes dihedral angles of 54.48 (12) and 57.91 (11)° with the pyridine and benzene rings, respectively. The morpholine ring atoms are disordered over two positions; the site-occupancy factors are ca 0.53 and 0.47. Inter-molecular C-H⋯F hydrogen ...

In the title compound, C(21)H(16)F(2)N(2), the seven-membered 1,4-diazepine ring of the benzodiazepine ring system adopts a distorted-boat conformation. The benzene ring of this system makes dihedral angles of 18.6 (2) and 78.8 (2)° with those of two fluoro-phenyl substituents. In the crystal, inversion dimers linked by two weak C-H⋯F hydrogen bonds generate R(2) (2)(20) ring motifs. There are ...

The title compound, C12H4Cl2F6N4OS {systematic name: 5-amino-1-[2,6-di-chloro-4-(tri-fluoro-meth-yl)phen-yl]-4-[(tri-fluoro-methane)sulfinyl]-1H-pyrazole-3-carbo-nitrile}, is a member of the phenyl-pyrazole group of acaricides, and one of the phenyl-pyrazole group of insecticides. The dihedral angle between the planes of the pyrazole and benzene rings is 89.03 (9)°. The fluorine atoms of the tr...

In the title mefloquine-oxazolidine derivative, C(24)H(20)F(6)N(2)O(2), the oxazoline ring adopts an envelope conformation (the flap atom is N) and the piperidine ring has a chair conformation. The oxazoline and benzene residues lie away from the C(6) ring of the quinoline group and, to a first approximation, to one side of the plane through the ten atoms (r.m.s. deviation = 0.025 Å). An intra-...

Throughout the text, let R be a commutative ring with identity (often called a commutative unital ring) and with multiplicative group of units R∗. Likewise, let f(x) = a0x +a1x n−1+ · · ·+an−1x+an be a polynomial of the variable x over R such that a0 ∈ R∗. Traditionally, R[x] is the ring of all polynomials of x over R; thereby f(x) ∈ R[x]. For an arbitrary but fixed element α, suppose f(x) is t...