let $r$ be an associative ring and let $m$ be a left $r$-module.let $spec_{r}(m)$ be the collection of all prime submodules of $m$ (equipped with classical zariski topology). there is a conjecture which says that every irreducible closed subset of $spec_{r}(m)$ has a generic point. in this article we give an affirmative answer to this conjecture and show that if $m$ has a noetherian spectrum, t...

This paper introduces two new algorithms for Lie algebras over finite fields and applies them to the investigate known simple of dimension at most $20$ field $\mathbb{F}_2$ with elements. The first algorithm is a approach towards construction $\mathbb{Z}_2$-gradings algebra characteristic $2$. Using this, we observe that each has $\mathbb{Z}_2$-grading determine associated superalgebras. second...

amara in jurisprudential sources is said to any evidence which indicate the fact. indication is not definitively in the sight of foghaha and osooliyeen, but due to the prevailing conjecture which can be achieved from that evidence. witnesses’ testimony (bayyinah) as valid evidence in the jurisprudence and law is known as an “amara” in the sight of jurists witch its validity is from legislator. ...

Two measures of how near an arbitrary function between groups is to being a homomorphism are considered. These have properties similar to conjugates and commutators. The authors show that there is a rich theory based on these structures, and that this theory can be used to unify disparate approaches such as group cohomology and the transfer and to prove theorems. The proof of the Schur-Zassenha...

Let $p$ be a prime, and $\mathbb{F}_p$ the field with elements. We prove that if $G$ is mild pro-$p$ group quadratic $\mathbb{F}_p$-cohomology algebra $H^\bullet(G,\mathbb{F}_p)$, then algebras $H^\bullet(G,\mathbb{F}_p)$ $\mathrm{gr}\mathbb{F}_p[\![G]\!]$ - latter being induced by quotients of consecutive terms $p$-Zassenhaus filtration are both Koszul, they quadratically dual to each other. C...

Let G be a subgroup of S,, given in terms of a generating set of permutations, and let p be a prime divisor of 1 G 1. If G is solvable-and, more generally, if the nonabelian composition factors of G are suitably restricted-it is shown that the following can be found in polynomial time: a Sylow p-subgroup of G containing a given p-subgroup, and an element of G conjugating a given Sylow p-subgrou...

Let GS be the Galois group of the maximal pro-p-extension QS of Q unramified outside a finite set S of places of Q not containing the prime p > 2. In this work, we develop a method to produce some examples of mild (and thus FAB) pro-p-group GS for which some relations are of degree 3 (according to the Zassenhaus filtration). The key computation are done in some Heisenberg extensions of Q of deg...