Let g be a simple Lie algebra over C. Fix a Borel subalgebra b and a Cartan subalgebra t ⊂ b. The corresponding set of positive (resp. simple) roots is ∆ (resp. Π). Write θ for the highest root in∆. An ideal of b is called ad-nilpotent, if it is contained in [b, b]. Consequently, such a subspace is fully determined by the corresponding set of positive roots, and this set is called an ideal (in ...

The reduction number r(A) of a standard graded algebra A is the least integer k such that there exists a minimal reduction J of the homogeneous maximal ideal m of A such that J m = m. Vasconcelos conjectured that r(R/I) ≤ r(R/ in(I)) where in(I) is the initial ideal of an ideal I in a polynomial ring R with respect to a term order. The goal of this note is to prove the conjecture. 1. Reduction ...

This research article is concerned with the introduction of a new notion normal spaces and regular spaces, namely mIαg-normal mIαg-regular spaces. We established their significant properties in ideal minimal Some equivalent conditions on are proved. Urysohn's Lemma also established.

It is shown that every dp-minimal integral domain R a local ring and for non-maximal prime ideal ℘ of R, the localization R℘ valuation ℘R℘ = ℘. Furthermore, if only its residue field infinite or finite maximal principal.

Starting from the constitutive properties that underpin the ideal memristor as originally defined by Leon Chua, we identify the conditions under which two memristors comply with the reciprocity theorem. In particular, we explore the minimal set of requirements for an ideal memristor and the physical implications of not complying with these criteria. Then, we show that reciprocity is satisfied w...

In the study of double-loop computer networks, the diagrams known as L-shapes arise as a graphical representation of an optimal routing for every graph’s node. The description of these diagrams provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each Cayley digraph with...

In this paper, it has been proved that for a Noetherian ring R and an automorphism σ of R, an associated prime ideal of R[x, σ] or R[x, x−1, σ] is the extension of its contraction to R and this contraction is the intersection of the orbit under σ of some associated prime ideal of R. The same statement is true for minimal prime ideals also. It has also been proved that for a Noetherian Q-algebra...

Argumentation Johannes P. Wallner, Georg Weissenbacher, and Stefan Woltran Institute of Information Systems, Vienna University of Technology, Favoritenstraße 9-11, A-1040 Vienna, Austria Abstract. In the area of propositional satisfiability (SAT), tremendous progress has been made in the last decade. Today’s SAT technology covers not only the standard SAT problem, but also extensions thereof, s...

We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has minimal reduction. When this is case, we study Cohen–Macaulay and Gorenstein properties associated graded provide several bounds for reduction number with respect to

The paper proposes a new construction for studying various properties of a polynomial with respect to an ideal generated by a finite basis of polynomials. These properties include, among others, (i) checking whether the polynomial is a zero divisor in the residue class ring defined by the associated ideal and (ii) checking whether the polynomial is invertible in the residue class ring defined b...