Let $R$ be a prime ring of characteristic not equal to $2$, $U$ the Utumi quotient and $C$ extended centroid $R$. $G$ $F$ two generalized derivations on $L$ non-central Lie ideal If $F\Big(G(u)\Big)u = G(u^{2})$ for all $u \in L$, then one following holds: (1) $G=0$. (2) There exist $p,q U$ such that $G(x)=p x$, $F(x)=qx$ $x R$ with $qp=p$. (3) satisfies $s_4$.

We introduce the notion of a Hu-Liu prime ideal in the context of left commutative rngs, and establish the contravariant functor from the category of left commutative rngs into the category of topological spaces. It is well known that new points must be introduced in order to expand algebraic geometry over algebraically closed fields into Grothendieck’s scheme theory over commutative rings. We ...

In the title compound, C(25)H(15)BrN(4), the two pyridine rings lie in a common plane [r.m.s. deviation = 0.023 (2) Å], whereas the bromo-phenyl and indole rings are twisted away from this plane by 52.82 (12) and 28.02 (10)°, respectively. The crystal structure is stabilized by inter-molecular N-H⋯N inter-actions.

We study the streamlines of ?-harmonic functions in planar convex rings. include polygons. The points where can meet are characterized: they lie on certain curves. gradient has constant norm along outside set meeting points, infinity-ridge.

The asymmetric unit of the title compound, C(8)H(8)BrI, contains three independent mol-ecules. In each molecule, the Br, I and C atoms of the methyl groups lie in the benzene ring plane. Intra-molecular C-H⋯I hydrogen bonds result in the formation of three planar five-membered rings, which are nearly coplanar with the adjacent rings.

Several results about the multiplicities of indecomposable injectives in the minimal injective resolution of a ring exist in the literature. Mostly these apply to universal enveloping algebras of finite dimensional solvable Lie algebras, and to Gorenstein noetherian PI local rings. We unify these results and extend them to the much wider class of rings with Auslander dualizing complexes.

We classify the spherical birational sheets in a complex simple simply-connected algebraic group. use classification to show that, when $G$ is connected reductive group with derived subgroup, two conjugacy classes $\mathcal{O}_1$, $\mathcal{O}_2$ of lie same sheet, up shift by central element $G$, if and only coordinate rings $\mathcal{O}_1$ are isomorphic as $G$-modules. As consequence, we pro...

Extending ideas of twisted equivariant K-theory, we construct twisted versions of the representation rings for Lie superalgebras and Lie supergroups, built from projective Z2-graded representations with a given cocycle. We then investigate the pullback and pushforward maps on these representation rings (and their completions) associated to homomorphisms of Lie superalgebras and Lie supergroups....

We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson–Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin–Drinfeld classification of Poisson–Lie structures on G corresponds to a cluster structure in O(G). We prove a reduction theorem explaining how different parts of the con...