Let $G$ be a group with identity $e$. $R$ commutative $G$-graded ring non-zero identity, $S\subseteq h(R)$ multiplicatively closed subset of and $M$ graded $R$-module. In this article, we introduce study the concept $S$-1-absorbing prime submodules. A submodule $N$ $(N:_{R}M)\cap S=\emptyset$ is said to prime, if there exists an $s_{g}\in S$ such that whenever $a_{h}b_{h'}m_{k}\in N$, then eith...

In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to groups (DGLG). We first recall classical problem integration in context, and recollect known results. Then, define category study its properties. show how associate a algebra every group vice versa. For DGLA ? DGLG direction, main “tools” are Hopf Harish-Chandra pairs (HCP)—we HCPs explain those rela...

groups of infinite rank in which every subgroup is either normal or self-normalizing are characterized in terms of their subgroups of infinite rank.

let $r=oplus_{nin bbb n_0}r_n$ be a noetherian homogeneous ring with local base ring $(r_0,frak{m}_0)$, $m$ and $n$ two finitely generated graded $r$-modules. let $t$ be the least integer such that $h^t_{r_+}(m,n)$ is not minimax. we prove that $h^j_{frak{m}_0r}(h^t_{r_+}(m,n))$ is artinian for $j=0,1$. also, we show that if ${rm cd}(r_{+},m,n)=2$ and $tin bbb n_0$, then $h^t_{frak{m}_0r}(h^2_{...

1. I n t r o d u c t i o n Graded normal logics ( G N L s ) are the extensions of modal normal logics to a language with graded modalities. The interpretation in usual Kripke models of a formula <>hA (n < w), whose main operator is a graded possibility, is there are more than n accessible worlds where A is true. When graded modalities were introduced (in [7], independently rediscovering a forme...

For any finitely generated abelian group Q, we reduce the problem of classification of Q-graded simple Lie algebras over an algebraically closed field of “good” characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional Q-graded simple Lie algebras over any algebraically closed field of characteristi...