Let R be a ring with identity.
The ain is to study some fundamental properties of a ring R when R is regular
or semi-potent and the radical Jacobson of R when R is semi-potent.
New results were obtained including necessary and sufficient condition
s of R
to be regular or semi-potent. New substructures of R are studied and their
relationship with the total of R.
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.
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 .
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.
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
conditi
ons the total of a ring R to equal some ideal of R.
