Locally Projective and Locally Injective Modules II

The object of this paper is to study the locally projective and locally injective modules. Specifically, this paper is a continuation of study of locally projective and locally injective modules, where a new description of locally projective and locally injective modules is obtained.

I -Semipotency and the Total of Modules

The object of this paper is to study the total as substructure of hom (M,N) R for any two modules R M and R N , one of interesting question, is when the total of a module N equals the hom (N, J (N)) R .

Locally Projective and Locally Injective Modules

The object of this paper is to study the endomorphism rings of locally projective and locally injective modules. Specifically, this paper is a continuation of study of endomorphism rings of locally projective and locally injective modules to be semipotent rings.

On (Δ−, ∇−, Ι−) semipotent and the total of rings and modules

Let M and N be two modules over a ring R. The object of this paper is the study of substructures of hom (M, N) R such as, radical, the singular, and co-singular ideal and the total. The new obtained results include necessary and sufficient conditions the total of a ring R to equal some ideal of R.

Maximal Elements and Prime Elements in Lattice Modules

In this paper we study the relationship between the maximal (prime) elements of M and the maximal (prime) elements of L. We show that, if L is a local lattice and the greatest element of M is weak principal, then M is local . Then we define the Jacobson radical of M and denote it by J(M) and we study its relationship with the Jacobson radical of L (J(L)) . Afterwards, we define the semiprime element in a lattice module M, and we show that the definitions of prime element and semiprime element are equivalent when the greatest element of M is multiplication and we study the properties equivalent to the properties of prime element in lattice module .

The semi-potent endomorphism ring of a module

In this paper, we studied the concept of semi-potency of endomorphism ring of modules. In addition to that, it has been studied the endomorphism ring of semi injective ( projective) modules and direct injective ( projective) modules.