Finitely Generated Modules over Group Rings of a Direct Product of Two Cyclic Groups

LetK be a commutative field of characteristic p > 0 and letG = G 1 ×G 2 , whereG 1 andG 2 are two finite cyclic groups.We give some structure results of finitely generated K[G]-modules in the case where the order of G is divisible by p. Extensions of modules are also investigated. Based on these extensions and in the same previous case, we show thatK[G]-modules satisfying some conditions have a fairly simple form.


Introduction
Let be a field of characteristic > 0 and let be a finite group. The study of [ ]-modules in the case where the order of is divisible by is a very difficult task. When is a finite abelian -group, we find in [1] the following statement: a complete classification of finitely generated [ ]-modules is available only when is cyclic or equal to 2 × 2 , where 2 is the cyclic group of order 2. In [2] we find this classification in these two cases. Still more, in the case where the Sylowsubgroup of is not cyclic, the groups such that = 2 and is dihedral, semidihedral, or generalized quaternion are the only groups for which we can (in principle) classify the indecomposable [ ]-modules (see [2]). These reasons just cited show the importance of the study of [ ]-modules when is of order divisible by and equal to a direct product of two cyclic groups. Now, let be a commutative field of characteristic > 0 and let = 1 × 2 , where 1 and 2 are two finite cyclic groups. Let be a finitely generated [ ]-module. When is considered as a module over a subalgebra [ ] of [ ] for a subgroup of the group , we write ↓ .
In Section 2, we show that if 1 is a cyclic -group and the characteristic of does not divide the order of 2 , then we can have a complete system of indecomposable pairwise nonisomorphic [ ]-modules. In the rest, we assume that 1 = ⟨ 1 ⟩ and 2 = ⟨ 2 ⟩ are cyclic -groups. Under conditions that ↓ 1 is a free [ 1 ]-module and that /( 1 − 1) is a free [ 2 ]-module, we show that is a free [ ]-module. We also show that if 2 is of order , ̸ = 0, and 2 is the subgroup of 2 generated by − 2 with 0 < ≤ , then under certain conditions is a free [ 1 × 2 ]-module. The fact that ↓ 1 must be a free [ 1 ]module is one of these conditions, and exactly in the end of this section we give a result that shows when this condition is satisfied. In Section 3 and always in the case where 1 and 2 are cyclic -groups, we show that under some conditions [ ]-modules have a fairly simple form. But in case = 2, 1 and 2 are two cyclic groups of respective orders 2 and 2 , ̸ = 0; these modules have this simple form without any other assumptions other than that they must be finitely generated over [ ].

Free [ × ]-Modules of Finite Rank
Throughout this paper, rings are assumed to be commutative with unity. We begin this section by giving a weak version of Nakayama's lemma with an elementary proof. (1) if ( − 1) = , then = 0;
Proof. (1) Let be the order of . We have is a natural number. So since has characteristic . Now = ( − 1) = ( − 1) 2 = ⋅ ⋅ ⋅ = ( − 1) = 0. (3) If is the submodule generated by , then = ( − 1) + , and then by (2) we have = . For a ring of prime characteristic and for a cyclic group of order generated by an element , we have the following lemma.

Proof. Define
: [ ] where is a well-defined -algebra homomorphism. It is easy to see that is surjective.  Let be a commutative field of characteristic > 0 and let be a direct product of two finite groups 1 and 2 . We Assume that 1 is a cyclic group of order generated by 1 and does not divide the order is a principal Artinian local ring with residue field (up to isomorphism) whose maximal ideal is generated by − 1. So is a principal Artinian local ring with residue field (up to isomorphism) whose maximal ideal is generated by 1 , where is a principal Artinian local ring of residue field . The characteristic of does not divide the order of 2 . Under these conditions, we can apply [3, Theorem 3.6] to have a complete system of indecomposable pairwise nonisomorphic [ ]-modules.
In the remainder of this section, we assume that 1 = and 2 = are two cyclic groups of respective orders and and are generated, respectively, by 1 Proof. This lemma is a particular case of a more general result (see [5, page 386]). But for this particular case, we can give the following direct proof: -module, and we have already seen Similarly we show that /( 2 − 1) is a [ 1 ]-module. Hence, As is a principal Artinian local ring with residue field and 1 − 1 is a generator of its maximal ideal.
So /( 1 −1) = 0. By Nakayama's lemma and the remark following it, = 0. Therefore, = which is projective Let be the Jacobson radical of [ ] for ∈ {1, 2}. Note that if is of characteristic (as here) and is a cyclicgroup, then the Jacobson radical of [ ] is none other than where is a generator of (see [5, page 122] [ ] where is a well-defined -algebra homomorphism. It is not difficult to show that is an isomorphism (using an argument similar to that done in the proof of Lemma 3). So / 1 is a free [ 2 ]-module.

Theorem 9.
Let be a [ ]-module of type , with 2 = 0, and let 2 be the subgroup of 2 generated by − 2 with 0 < ≤ . If ↓ 1 is -free and = , then is a free
(2) ⇒ (3) Now, assume that dim ( ) = ( /( − 1))dim (( −1) ). So dim ( ) = ×(dim ( )−dim (( − 1) )). As dim ( )−dim (( −1) ) is the number of blocks of the Jordan matrix of , the order of each block is less than or equal to , and dim ( ) is equal to the sum of the orders of these blocks, then the order of each block is . Therefore is equal to the number of Jordan blocks of . Therefore the order of each Jordan block is equal to . So the modules contained in a decomposition of ↓ ⟨ ⟩ as a direct sum of indecomposable modules are of the form [⟨ ⟩]; that is, ↓ ⟨ ⟩ is free.

Use of Module Extensions
Let be a finite group and let be a ring. Let 1 and 2 be two [ ]-modules. We put = Hom ( 2 , 1 ). has a natural structure as a ( [ ], [ ])-bimodule centralized by (see [7, section 25]). Explicitly, we have A derivation : [ ] → is an -homomorphism satisfying This [ ]-module is denoted by 1 × 1 as in [8]. An extension of 2 by 1 is an [ ]-exact sequence be a pair of extensions of 2 by 1 . These two extensions are equivalent if there exists an isomorphism of [ ]-modules Φ : → such that Φ = and V Φ = V. These equivalence classes of extensions form an -module where and denote, respectively, the canonical injection from 1 to 1 × 2 and the second projection from 1 × 2 to 2 , is exact. The equivalence class of this sequence is denoted by Remark 11. With the previous notations, derivations and modules 1 × 2 play the same role as the cocycles and modules 1 × 2 defined in [8].
From Proposition 25.10 of [7] we have the following result. Proof. Let : → be an isomorphism of [ 2 ]modules. We extend the action of 2 on to by ⋅ = , ∀( , ) ∈ 1 × . We easily see that the application : → is an isomorphism of [ ]-modules.