The objective of this paper is to introduce the concept refined neutrosophic matrices as such multiplication, addition, and ring property. Also, it determines necessary sufficient condition for invertibility these with respect multiplication. On contrary, nilpotency idempotency properties will be discussed.

Abstract We discuss the theory of Lie algebras in Lean’s Mathlib library. Using nilpotency as theme, we outline a computer formalisation Engel’s theorem and an application to root space theory. emphasise that all arguments work with coefficients any commutative ring.

In this work we study Leibniz algebras whose second-maximal subalgebras are ideals. We provide classification based on solvability, nilpotency, and properties of the Frattini subalgebra. give several specific descriptions with these include necessary sufficient conditions for non-nilpotent case.

Let G be a finite p-group. We prove that whenever the commuting probability of G is greater than (2p2 + p− 2)/p5, the unramified Brauer group of the field of G-invariant functions is trivial. Equivalently, all relations between commutators in G are consequences of some universal ones. The bound is best possible, and gives a global lower bound of 1/4 for all finite groups. The result is attained...

Abstract For the $p$ -localized sphere $\mathbb {S}^{2m-1}_{(p)}$ with $p >3$ a prime, we prove that homotopy nilpotency satisfies $\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}<\infty$ , respect to any associative $H$ -structure on . We also {S}^{2m-1}_{(p)}= 1$ for all but finite number of primes Then, loop space associated -projective {S}^{2m-1}_{(p)}P(n-1)$ $m,n\ge 2$ and $m\mid p-1$ derive \...