In this paper, we introduce nilpotent soft (sub)polygroups. In addition, nilpotency of intersection, extended intersection, restricted union of two nilpotent soft polygroups are studied. Espesialy, a necessary and suficient condition between nilpotency of a polygroup and soft polygroups is obtained. Finally, we define two new soft polygroups (Sα )A∪ {c} and (Qα )A derived from a soft polygroup α A and study on nilpotency of these structures. Also, we extend a soft homomorphism of groups to polygroups. This helps us to extend some properties of groups to polygroups.

We introduce the notion of nilpotent p.p. rings, and prove that the nilpotent p.p. condition is preserved over polynomial rings and skew polynomial rings.

Applications of hypergroups have mainly appeared in special subclasses. One of the important subclasses is the class of polygroups. In this paper, we study the notions of nilpotent and solvable polygroups by using the notion of heart of a polygroup.In particular, we give a necessary and sufficient condition between nilpotency (solvability) of polygroups and fundamental groups.

Let n = [x1, : : :, xn] be the nth lower central word. Suppose that G is a pro, nite group where the conjugacy classes x n(G) contains less than 2@ 0 elements for any x 2 G. We prove that then n+1(G) has , nite order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is , nite. Moreover, it implies that a pro, nite group G is , nite-by-nilpotent if and only if there is a positive integer n such that x n(G) contains less than 2@ 0 elements, for any x 2 G.

In this article, using the definitions of central series and nilpotency in the Lie algebras, we give some results similar to the works of Hulse and Lennox in 1976 and Hekster in 1986. Finally we will prove that every non trivial ideal of a nilpotent Lie algebra nontrivially intersects with the centre of Lie algebra, which is similar to Philip Hall's result in the group theory.

Let $L$ be a finite dimensional Lie algebra over an arbitrary field $F$. In this paper, we prove that the class of finite nilpotent(solvable) Lie algebras is an example of formation. Furthermore, we conclude that every finite Lie algebra has a nilpotent(solvable) residual. Finally we prove some results on Frattini and Fitting subalgebras of the nilpotent Lie algebra $L$.

Let R be a ring with an endomorphism  and an -derivation . Antoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. In this paper we introduce and investigate the notion of nil-( ,  )-compatible rings. The class of nil-( ,  )-compatible rings are extended through various ring extensions and many classes of nil-( ,  )-compatible rings are constructed. We also prove that, if R is nil--compatible and nil-Armendariz ring of power series type with nil  R  nilpotent, then nil(R[[x;  ]])  nil(R)[[x;  ]]. We show that, if R is a nil-Armendariz ring of power series type, with nil  R  nilpotent and nil-( ,  )-compatible ring, then nil  R  x;  ,     nil  R   x;  ,   . As a consequence, several known results are unified and extended to the more general setting. Also examples are provided to illustrate our results.

Fundamental relations are one of the main tools in connection relation between hyperstructures theory and structures theory. In this paper, we introduce a general fundamental relation on any hypergroup in such a way that all fundamental relations are a special case of this relation. Also, this study considered the notation of the relation on the derivation of k-nilpotent groups from any hypergroups.