M, N:M denotes the subset, in fact an ideal of R, consisting of all elements x  R such that x  M  N. 

For a non-empty multipicatively closed subset of S of R and an R-submodule N of M, N(S) denotes the 

R-submodule of M consisting of all elements x  M so that sx  N for at least one s  S. M_S and R_S 

denotes the localization of M and R at S respectively. 

            An R-submodule N of M is called a prime R-submodule of M if and only if  N # M and whenever 

xx  N, for x  R and x  M , we have either x  N or whenever xx  N, for x  R and x M, we 

have either x  N or x  rad( N : M ). These definitions do not put any further restriction neither on 

prime ideals nor on primary ideals in case M-R. 

            Several studies have been carried out by E.H.Feller, H.Marubayashi, D.G.Northcott and 

E.W.Swokowski to develop a module theoretical analogue of the ideal theory. But in some way or 

another their study is not complete. It is the purpose of this thesis, using the above definition of prime 

-Rsubmodules, to do a complete module theoretical analogy of prime ideal theory. 

            General properties of prime R-submodules as the existence of minimal prime R-submodules over 

proper R-submodules are obtained and a generalization of Cohen's for rings is stated for modules. The 

one-one correspondence between the set of prime R-submodules N of M such that (N : M ) Ç S -  f  

and the set of all prime R_S-submodules of M_S is proved. We say that two prime R-submodules N₁ and 

N₂ are related, and write N₁ ~ N₂ if and only if N₁:M-N₂:M. This equivalence relation partitions the set of 

all prime R-submodules of M into disjoint classes. Since N₁:M is a prime ideal of R. Denote each class by 

C_p where P-N:M for each N in this class. It is proved that PM(S), where S - R \ P , is the unique minimal 

element in the class C_p. This minimal element is called a distinguished prime R-submodule of M belonging 

to the prime ideal P. The one-one correspondence between the set of all prime ideals of R containing 

Ann_R M, where M is finitely generated, and the set of all distinguished prime R-submodules of M is 

proved. Similar concepts as distinguished primary submodule and minimal R-submodule belonging to an 

ideal of R are introduced and related properties are proved. Using distinguished prime R-submodules, the 

notions of radical, rank, corank, and dimension for rings are extended to modules. The relation of this 

definition of dimension to other definition is investigated. 

             In this way most of the ideas in ideal theory can be interpreted for modules.