Let F be a free Lie algebra of finite rank over a field K. We prove that if an ideal [Formula: see text] of the algebra [Formula: see text] contains a primitive element [Formula: see text] then the element [Formula: see text] is primitive. We also show that, in the Lie algebra [Formula: see text] there exists an element [Formula: see text] such that the ideal [Formula: see text] contains a prim...

We show that in a local S1 ring every two-generated ideal of linear type can be generated by a two-element sequence of linear type and give an example which illustrates that the S1 condition is essential. We also show that every Noetherian local ring in which every two-element sequence is of linear type is an integrally closed integral domain and every two-generated ideal of it can be generated...

Reversible computing is a concept reflecting physical reversibility. Until now several reversible systems have been investigated. In a series of papers Kenichi Morita defines the rotary element RE, that is a reversible logic element. By reversibility, he understands [2] that ’every computation process can be traced backward uniquely from the end to the start. In other words, they are backward d...

In this paper, a new derivation for one of the black hole line elements is given since the basic derivation for this line element is flawed mathematically. This derivation postulates a transformation procedure that utilizes a transformation function that is modeled by an ideal nonstandard physical world transformation process that yields a connection between an exterior Schwarzschild line eleme...

Proposition 0.1. Let R be a commutative ring. TFAE: 1. Every ideal in R is finitely generated. 2. Every chain of ideals I1 ⊆ I2 ⊆ · · · is stable. That is, there exists n such that In = In+1 = · · · . 3. Every nonempty set of ideals has a maximal element wrt inclusion. Proof. 1. (1) =⇒ (2). I = ⋃ k Ik ideal in R. I is generated by x1, x2, . . . xm. There exists n such that x1, . . . , xm ∈ In. ...

Let $R$ be an associative ring with unity. An element $x \in R$ is called $\mathbb{Z}G$-clean if $x=e+r$, where $e$ is an idempotent and $r$ is a $\mathbb{Z}G$-regular element in $R$. A ring $R$ is called $\mathbb{Z}G$-clean if every element of $R$ is $\mathbb{Z}G$-clean. In this paper, we show that in an abelian $\mathbb{Z}G$-regular ring $R$, the $Nil(R)$ is a two-sided ideal of $R$ and $\fra...

in this paper a particular case of z-ideals, called strongly z-ideal, is defined by introducing zero sets in pointfree topology. we study strongly z-ideals, their relation with z-ideals and the role of spatiality in this relation. for strongly z-ideals, we analyze prime ideals using the concept of zero sets. moreover, it is proven that the intersection of all zero sets of a prime ideal of c(l),...

Let k be a field and let A = C[x1, . . . , xn]. For the set of common zeros of elements of an ideal I ⊆ A, we write V(I). For the ideal of polynomials which vanish on a set V ⊆ k, we write I(V ). Given an ideal I ⊆ A, the ideal membership problem is the problem of determining whether or not a given f ∈ A is an element of the ideal I. This is difficult because the division algorithm can badly fa...

A finite lattice is interval dismantlable if it can be partitioned into an ideal and a filter, each of which can be partitioned into an ideal and a filter, etc., until you reach 1-element lattices. In this note, we find a quasi-equational basis for the pseudoquasivariety of interval dismantlable lattices, and show that there are infinitely many minimal interval non-dismantlable lattices. Define...