Let N be a submodule of a module M and a minimal primary decomposition of N is known. A formula to compute Baer's lower nilradical of N is given. The relations between classical prime submodules and their nilradicals are investigated. Some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated.

The notion of multiplicative hyperrings is an important class of algebraic hyperstructures which generalize rings where the multiplication is a hyperoperation, while the addition is an operation. Let R be a commutative multiplicative hyperring and α,∈,End(R). A proper hyperideal I of R is called α,-prime if x ◦,y ⊆,I for some x, y ∈,R then x ∈,I or α, (y) ∈,I. Indeed, the α,-prime hyperideals are a new generalization of prime hyperideals. In this paper, we aim to study α,-prime hyperideals and give the basic properties of this new type of hyperideals.

The main result of our consideration is the proof of the centrally nilpotency of class two property for connected topological proper loops L of dimension  3 which have an at most six-dimensional solvable indecomposable Lie group as their multiplication group. This theorem is obtained from our previous classification by the investigation of six-dimensional indecomposable solvable multiplication Lie groups having a five-dimensional nilradical. We determine the Lie algebras of these multiplication groups and the subalgebras of the corresponding inner mapping groups.

Let R be a commutative ring with identity. We give a new generalization to prime ideals called ,-prime ideal. A proper ideal P of R is called an ,-prime ideal if for all a,b in R with ab 2 P, then a 2 P or , (b) 2 P where ,2 End(R). We study some properties of ,-prime ideals analogous to prime ideals. We give some characterizations for such generalization and we prove that the intersection of all ,-primes in a ring R is the set of all ,-nilpotent elements in R. Finally, we give new versions of some famous theorems about prime ideals including ,-integral domains and ,-, elds.