A celebrated result of Herstein [10, Theorem 6] states that a ring R must be commutative if[x,y]n(x,y)=[x,y] for all x, y ∈ R, wheren (x,y)>1 is an integer. In this paper, we investigate the structure prime satisfies identity F([x,y])n=F([x,y]) and σ([x,y])n=σ([x,y]), where F σ are generalized derivation automorphism respectively n>1a fixed

In the combination of 5-fluorophenyl-1, 3, 7, 9-tetra methylpyrido [2, 3, d, d, 5, 6 dipyrimidine -2, 4, 6, and 8 tetron, there are 4 rings, respectively, ring that is attached to the halogen ring A and, respectively, a ring The lower and the right to the left of the ring A are called C, B, and D. Calculations and studies show that the amount of aromatics in halogen-ring A in all cases has the ...

In the title compound, C(11)H(10)N(3)O(4)S(+)·C(2)F(3)O(2) (-), the benzene ring makes an angle of 87.3 (2)° with the pyridinium ring. The nitro group is essentially coplanar with the benzene ring. The F atoms of the CF(3) group are disordered over two positions with almost equal occupancy [0.531 (12)/0.469 (12)]. The crystal structure is stabilized by N-H⋯O and C-H⋯O hydrogen bonds.

A construction due to Knörrer shows that if N is a maximal Cohen–Macaulay module over hypersurface defined by f+y2, then the first syzygy of N/yN decomposes as direct sum and its own syzygy. This was extended Herzog–Popescu hypersurfaces f+yn, replacing N/yn−1N. We show, in same setting Herzog–Popescu, N/ykN always an extension syzygy, moreover this has useful approximation properties. give two...

Let X be an m × n matrix (m ≤ n) of indeterminates over a field K of positive characteristic, and denote the ideal generated by its t-minors by It . We show that the Rees ringR(It ) ofK[X], as well as the algebra At generated by the t-minors, are F-rational if charK > min(t, m − t). Without a restriction on characteristic this holds for K[X]/Ir+1 and the symbolic Rees ring R(It ). The determina...

in this article, we study the artin-rees property in  c(x), in the  rings of fractions of  c(x) and in the factor rings of c(x) . we show that c(x)/(f) is an artin-rees ring if and only if  z(f)  is an open p-space. a necessary and sufficient condition for the local rings of  c(x)   to be artin-rees rings is that each prime ideal in  c(x)  becomes minimal and it turns out that every local ring ...

There are many instances of the principle that if A is an algebra of functions on X, then every ring homomorphism A → C is given by evaluation at a particular x ∈ X. Examples are the nullstellensatz in algebraic geometry and the result of Gelfand when X is a compact Hausdorff space. In these cases one can regard X as being included in Hom(A, C) as the set of those f : A → C which satisfy the se...

Let $f(x)\in\mathbb{Z}[x]$ be a nonconstant polynomial. $n, k$ and $c$ integers such that $n\ge 1$ $k\ge 2$. An integer $a$ is called an $f$-exunit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(f(a),n)=1$. In this paper, we use principle cross-classification to derive explicit formula for number ${\mathcal N}_{k,f,c}(n)$ solutions $(x_1,...,x_k)$ congruence $x_1+...+x_k\equi...

In the title compound, C(13)H(9)F(2)N(3)O, the pyridine ring forms a dihedral angle of 16.92 (7)° with the benzene ring. In the crystal, mol-ecules are linked via N-H⋯O, C-H⋯O and C-H⋯F, with the same O atom accepting two bonds.