On Submodules of Modules over Group Rings

. In this paper, we ﬁnd some connections between submodules of a module over a group ring RG and subgroups of a group G . Also, we prove that there is a direct connection between conjugate elements of G and RG -submodules of M . Finally, we show that there is a correspondence between the associative powers (cid:52) iM ( G ) of (cid:52) M ( G ) and i th dimension subgroups (cid:53) ( (cid:52) iR ( G )) of G over R .


Introduction
The history of group rings dates back to long time and since then, many survey articles have appeared ( [3], [6], [12], [13], [17]). But modules over group rings are one of the subjects studied in recent years by a lot of authors interested in algebra ( [4], [5], [7], [8], [16]). As distinct from the definition of group module over group rings as defined in [7], we gave a definition of group module over group rings with the help of a group homomorphism from G to End(M) in [16]. As a continuous study of our previous paper, in here by this previous definition, we give some characterizations for modules over group rings in this paper.
Throughout this paper, all rings are commutative with identity and all modules are unital group modules over group rings unless stated otherwise.
Let R be a commutative ring with identity and G a group. The group ring of G over R is denoted by RG, which is the set of all formal expressions of the form ∈G r where r ∈ R and r = 0 for almost every ∈ G.
For elements r = ∈G r , s = ∈G s ∈ RG, by writing r = s, we mean r = s for all ∈ G.
In fact, RG is a ring with the following sum and multiplication The augmentation map of RG is a ring homomorphism from RG to R given by ∈G r → ∈G r and its kernel denoted by R (G) is called the augmentation ideal of RG. In other words, the ideal R (G) of RG is defined as the following set: For a left ideal I of RG, (I), which is a subgroup of G, is defined as the following set: One can observe that ( R (G, H)) = H for a subgroup H of G. Based on the definition and structure of a group ring, a group module over a group ring is defined as follows: Let τ be a group homomorphism from G to End(M). For all ∈ G, m ∈ M, the multiplication m is defined as In here, M is an RG-module with this multiplication and the group homomorphism τ is a representation of G for M over R ( [16]).
Alkan generalized the augmentation mapping of RG to the group module in [2] and proved the following property: where H is a normal submodule of G. Besides the relations among RG-submodules of a group module MG with regard to normal subgroups and elements of G, there are other relations among subgroups of G and R (G), its associative powers i R (G) and M (G), its associative powers i M (G) which are defined below. Firstly recall that associative powers . is a filtration of the augmentation ideal R (G). This filtration has the property that i R (G).
. In this paper, we find some connections between RG-submodules of M over RG and subgroups of G and a correspondence between associative powers i M (G)s of M (G) and ith dimension subgroups ( i R (G)) of G.
In Section 2, we firstly deal with the structure of RG-submodules related to subgroups of G in Lemma 2.1 and Proposition 2.2. After giving a relation for a normal group of G related to an RG-submodule in Lemma 2.4, we prove that Before Theorem 2.6, we deal with an RG-submodule of M to find a correspondence between conjugate elements of G and RG-submodules of M. Thus we close Section 2 by the following result: If where ∈ G and H i is a subgroup of G.
In Section 3, we firstly define associative powers of 3. Finally, in Theorem 3.5, we prove that there is a correspondence between the descending module filtration

and the filtration
of G.

Submodules related to normal subgroups
In this section, we examine the connections between RG-submodules of M over RG and subgroups of G.   ii) Let x be in ΣN i (H). Then x = h∈H n h (h−1) with n h ∈ N i . Since n h ∈ N i , we have n h = t 1 h +t 2 h +...+t k h with t i h ∈ N i , and so This completes the proof.

Corollary 2.3. Let N be an RG-submodule of an RG-module M. If N is generated by the set S, then we have
Proof. The proof of this result is similar to the proof of (ii) in Proposition 2.2.
Let N be an RG-submodule of M. Then a subgroup (N) of G is defined in [2] as Then we give the following lemma. Proof. i) Using the distributive law, we have the following: Let be in ∩H i . Then ∈ H i for each i. Since H i = ( M (G, H i )), it follows that ∈ ( M (G, H i )).
Theorem 2.5. Let x i ∈ G and N be an RG-submodule of an RG-module M. If (N) =< x 1 , x 2 , ..., x k >, then we have ., x k >, for each x i ∈ (N), there exist some positive numbers y 1 h , y 2 h , ..., y k h such that h = x We recall a well-known definition from [9]. Let and h be two elements of a group G. The element h −1 is called the conjugate of h by .
We are now ready to prove the relation between conjugate subgroups of G and RG-submodules of M. Since Since m h r k ∈ M, it follows xy ∈ M.(K − 1)(H − 1). To prove its converse, the same method can be used.
ii) It is enough to show M.
Then there exist m ∈ M and t ∈ RG((H ∩ K) − 1) 2 such that x = mt. Also there exist a i , b i ∈ (H ∩ K) and r i ∈ RG such that t = r i (a i − 1)(b i − 1). Thus we have To prove its converse, the same method can be used. of G. In other words, there is a correspondence between the associative powers i M (G) of M (G) and ith dimension subgroups ( i R (G)) of G over R.
Proof. We have a correspondence between i M (G) and i R (G) by the equality i M (G) = M. i R (G) given above. For all i R (G) are ideals of RG, we have another correspondence between the associative powers i R (G) of R (G) and the ith dimension subgroups of G over R via G ∩ (1 + i R (G)) = ( i R (G)). Consequently, we have the correspondence between i M (G) and ( i R (G)). Hence, we get the following diagram which shows the desired correspondence between the descending module filtration of M (G) and the filtration of G by the ith dimension subgroups of G over R.