Recent studies have shown that hypergraphs are useful in solving real-life problems. Hypergraphs have been successfully applied in various fields. Inspiring by the importance, we shall introduce a new hypergraph assigned to a given module. By the way, vertices of this hypergraph (which we call sum hypergraph) are all nontrivial submodules of a module $P$ and a subset $E$ of the vertices is a hyperedge in case the sum of each two elements of $E$ is equal to $P$ and $E$ is maximal with respect to this condition. Some general properties of such hypergraphs are discussed. Semisimple modules with length $2$ are characterized by their corresponding sum hypergraphs. It is shown that the sum hypergraph assigned to a finite module $P$ is connected if and only if $P$ is semisimple.

We give some new characterizations of semisimple alternative algebras among the pseudo-Euclidean alternative algebras. These charcterizations are based on the index of a pseudo Euclidean alternative algebra, operators of Casimir type and representations of alternative algebras.

Since the theory of filters plays an important role in the theory of lattices, in this paper, we will make an intensive study of the notions of semisimple lattices and the socle of lattices based on their filters. The bulk of this paper is devoted to stating and proving analogues to several well-known theorems in the theory of the rings. It is shown that, if $L$ is a semisimple distributive lattice, then $L$ is finite. Also, an application of the results of this paper is given. It is shown that if $R$ is a right distributive ring, then the lattice of right ideals of $R$ is semisimple iff  $R$ is a semisimple ring.

We show that every semi-artinian module which is contained in a direct sum of finitely presented modules in s [M], is weakly co-semisimple if and only if it is regular in s [M]. As a consequence, we observe that every semi-artinian ring is regular in the sense of von Neumann if and only if its simple modules are FP-injective.

We investigate and generalize quantum determinants to semisimple spherical and pivotal categories. It is well known that traces are preserved by strong tensor functors; we show on one hand that in fact, weaker conditions on a functor are sufficient to continue preserving traces. On the other hand, we prove that these determinants are well-behaved under strong tensor functors. Further, we introduce a notion of domination rank for objects of a semisimple pivotal category and prove similar properties of the ordinary case. Furthermore, we expand the determinantal and McCoy ranks to introduce a morphism quantum rank function on a semisimple pivotal category.

In this paper, we define prime (semiprime) hyperideals and prime (semiprime) fuzzy hyperideals of semihypergroups. We characterize semihypergroups in terms of their prime (semiprime) hyperideals and prime (semiprime) fuzzy hyperideals.

Let $R$ be a commutative ring with identity and $M$ be an $R$-module. It is shown that the usual lattice $\mathcal{V}(_{R}M)$ of varieties of submodules of $M$ is a distributive lattice. If $M$ is a semisimple $R$-module and the unary operation $^{\prime}$ on $\mathcal{V}(_{R}M)$ is defined by $(V(N))^{\prime}=V(\tilde{N})$, where $M=N\oplus \tilde{N}$, then the lattice $\mathcal{V}(_{R}M)$ with $^{\prime}$ forms a Boolean algebra. In this paper, we examine the properties of certain mappings between $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$, in particular considering when these mappings are lattice homomorphisms. It is shown that if $M$ is a faithful primeful $R$-module, then $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$ are isomorphic lattices, and therefore $\mathcal{V}(_{R}M)$ and the lattice $\mathcal{R}(R)$ of radical ideals of $R$ are anti-isomorphic lattices. Moreover, if $R$ is a semisimple ring, then $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$ are isomorphic Boolean algebras, and therefore $\mathcal{V}(_{R}M)$ and $\mathcal{L}(R)$ are anti-isomorphic Boolean algebras.

manifold with a smooth action of a Lie group G is called G-manifold. In this paper we consider a complete Riemannian manifold M with the action of a closed and connected Lie subgroup G of the isometries. The dimension of the orbit space is called the cohomogeneity of the action. Manifolds having actions of cohomogeneity zero are called homogeneous. A classic theorem about Riemannian manifolds of nonpositive curvature states that a homogeneous Riemannian manifold M of nonpositive curvature is diffeomorphic to Rk ×Tm, dim M=m+k. There are many interesting theorems about topological properties of cohomogeneity one G-manifolds of nonpositive curvature. It is proved that if M is a negatively curved cohomogeneity one Riemannian G-manifold and dim M>3, then either M is diffeomorphic to, Rk ×Tm, dim M=m+k or p1 (M) =Z.Also, we know that if M is a G –manifold of negative curvature and G is semisimple, then M is simply…