Let L be a Lie algebra with the universal enveloping algebra U(L). The augmentation ideal ω(L) of U(L) is the associative ideal of U(L) generated by L. Let S be a subalgebra of L and n a positive integer. In this paper we prove that L ∩ ω(L)ω(S) = γn+2(S) + γn+1(S∩γ2(L)), where γn+2(S) is the (n+2) term of the lower central series of S.

Let L be a free restricted Lie algebra and R a restricted ideal of L. Denote by u(L) the restricted enveloping algebra of L and by ω(L) the associative ideal of u(L) generated by L. The purpose of this paper is to identify the subalgebra R ∩ ω(L)ω(R) in terms of R only . This problem is the analogue of the Fox problem for free groups.

We have already learned ideal of a ring. A lattice is an algebraic structure which looks like a ring for both have two operations. Similarly, a lattice also has a special structure named as the same as ideal. Definition 1. A subset I of a lattice L is an ideal if it is a sublattice of L and x ∈ I and a ∈ L imply that x ∩ a ∈ I. Specially, A proper ideal I of L is prime if a, b ∈ L and a ∩ b ∈ I...

Gu map-1 is a modified version of GGH map. It uses same ideal lattices for constructing the trapdoors, while the novelty is that no encodings of zero are given. In this short paper we show that Gu map-1 cannot be used for the instance of witness encryption (WE) based on the hardness of 3-exact cover problem. That is, if Gu map-1 is used for such instance, we can break it by soving a combined 3-...

let $l$ be a completely regular frame and $mathcal{r}l$ be the ringof continuous real-valued functions on $l$. we study the frame$mathfrak{o}(min(mathcal{r}l))$ of minimal prime ideals of$mathcal{r}l$ in relation to $beta l$. for $iinbeta l$, denoteby $textit{textbf{o}}^i$ the ideal${alphainmathcal{r}lmidcozalphain i}$ of $mathcal{r}l$. weshow that sending $i$ to the set of minimal prime ideals...

We extend the basic fact that every ideal of a finite dimensional semisimple Lie algebra has a unique complement to the case of closed ideals of prosemisimple Lie algebras. We prove that if A is a closed ideal of a prosemisimple Lie algebra L = lim ←−−Ln (n ∈ N), where the Ln are finite dimensional semisimple Lie algebras, then there exists a unique ideal B of L such that L = A ⊕ B. Mathematics...

Let $L$ be a completely regular frame and $mathcal{R}L$ be the ring of continuous real-valued functions on $L$. We study the frame $mathfrak{O}(Min(mathcal{R}L))$ of minimal prime ideals of $mathcal{R}L$ in relation to $beta L$. For $Iinbeta L$, denote by $textit{textbf{O}}^I$ the ideal ${alphainmathcal{R}Lmidcozalphain I}$ of $mathcal{R}L$. We show that sending $I$ to the set of minimal prime ...

it is called nontrivial if {0} 6= J 6= X. This notion was introduced by Alfsen and Effros [1] and has proved useful in Banach space geometry, approximation theory and harmonic analysis; see [12] for a detailed account. A number of authors have studied the M -ideal structure in L(X), the space of bounded linear operators on a Banach space X, with special emphasis on the question whether K(X), th...

We show that the cofinalities of both the Miller ideal m (the σ-ideal naturally related to Miller forcing M) and the Laver ideal l (related to Laver forcing L) are larger than the size of the continuum c in ZFC.