Let G be a group, Ω a set and K = {g ∈ G | ω * g = ω, Ɐω ∈ Ω} the nucleus of Ω where G acts on the set Ω. We will show that G/K is simple in the case that the group G verifies to be primitive on Ω, as well as that it is equal to its derived subgroup and finally if α ∈ Ω then Gα has a subgroup A that is abelian and normal such that G =< Ag | g ∈ G >, where Gα is the stabilizer of α in G. To finish we will give an application that the alternating group A5 is simple.

Daremos un ejemplo de un dominio de integridad que posee elementos no nulos con infinitos divisores primos.

Let p a prime number. The most familiar construction of the ring of p-adic integers ℤp, is as the projective limit of quotients of powers of the ideal (p)◁ℤ. There is another description of ℤp as a quotient of the power series ring ℤ[[X]], which can be found in some texts of p-adic analysis (see e.g. [3]). More specifically, there exists a ring isomorphism. Ψ : ℤ[[X]]/〈p − X〉 → ℤp. However, this isomorphism is also topological in nature, but there is no proof of this fact in the corresponding literature. In this article we will prove in sufficient detail that the above description is also valid in the context of topological rings.