A left ideal I of a ring R called a left principal annihilator if I = l(a) ={r∈R| ra =۰} for some a∈R. A ring R called left morphic if for all a∈R there exists b∈R such that Ra=l(b), Rb=l(a). Our interest here is in the ring R satisfying : for all a∈R there exist b∈R such that Ra=l(b). these ring were called left pseudo-morphic by Yang.
We show left pseudo-morphic rings are right p-injective. Also semiprime left pseudo-morphic rings are emismile and studied condition ACC on pseudo-morphic rings. We will show left pseudo-morphic rings, mininjective, with the ACC on {l(b)|b∈R}, are quasi-Frobenius and every right ideal is principal.
 

A submodule N ≠ M of an R-module M is said to be prime if axϵN, aϵR, xϵM\N implies aϵ(N:_RM) or equivalently , (N:_RM)=P is prime and M/N is torsion-free over R/P
A proper submodule N of an R-module M to be strongly prime if ((N+Rx):RM)y ⊆ N implies xϵN or yϵN for x , y ϵ M 
An integral domain R with quotient field K is said to be a G-domain if K=R[u^-1] for some nonzero uϵR and an ideal P of a ring R is called a G-ideal if R/P is a G-domain. We define a submodule N of a finitely generated R-module M to be a G-submodule of M if P =(N:RM) is a G-ideal of R and N is P-maximal
Strongly prime submodules of a finitely generated R-module are characterized
G-submodules are defined and it is shown that any prime submodule of a finitely generated R-module is an intersection of G-submodules of M. A finitely generated R-module M is defined to be a Jacobson module if each G-submodule of M is maximal .It is also shown that every finitely generated module over a Jacobson ring is Jacobson. Finally, the strongly prime submodules of a module are compared to the cocritical submodules which were defined by Nishitani

Th roughout this R denotes a commutative ring with identity and all

modules are unital R-modules We characterize Q-modules and almost

Q-modules. Next we establish some Equivalent conditions for an

almost Q-module to be a Q-module. Using these results some

characterizations are given for Noetherian Q-modules.

 

 

 The results of Szele and Szenderi [‘On Abelian groups with commutative en-
domorphism rings’, Acta Math. Acad. Sci. Hungar. 2 (1951), 309-324] charac-
terizing abelian groups with commutative endomorphism rings are generalized
to modules whose endomorphism rings have variouse restrictions on their idem-
potents. Such properties include central or commuting idempotents, and one-
sided ideals being two-sided. Related properties include direct summands having
unique complements, or being fully invariant.

In this paper, firstly survey multiplication module in chapter tow and then several properties of endomorphism rings of modules are investigated. A multiplication module M over a commutative ring R induces a commutative ring M^*of endomorphisms of M and hencethe relation between the prime (maximal) submodules of M and the prime (maximal) ideals of M^*can be found. In particular, two classes of ideals of M^*are discussed in this paper

we call supplemented self projective modules strongly discrete modules. we characterize discrete and strongly modules in terms of lifting of maps and prove some of their properties.we show that epi projective lifting modules are preciesly the strongly discrete modules and dually mono injective extending modules are preciesly the self injective modules.we also prove that vN-injective modules are preciesly the self injective modules