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

Let be a commutative field of characteristic and let , where and are two finite cyclic groups. We give some structure results of finitely generated -modules in the case where the order of is divisible by . Extensions of modules are also investigated. Based on these extensions and in the same previous case, we show that -modules satisfying some conditions have a fairly simple form. 1. Introduction Let be a field of characteristic 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 , where is the cyclic group of order 2. In [2] we find this classification in these two cases. Still more, in the case where the Sylow -subgroup of is not cyclic, the groups such that 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 and let , where and 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 is a cyclic -group and the characteristic of does not divide the order of , then we can have a complete system of indecomposable pairwise nonisomorphic -modules. In the rest, we assume that and are cyclic -groups. Under conditions that is a free -module and that is a free -module, we show that is a free -module. We also show that if is of order , , and is the subgroup of generated by with , then under certain conditions is a free -module. The fact that must be a free -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 and are cyclic -groups, we show that under some conditions -modules have a fairly simple form. But in case , and are two cyclic groups of respective orders and , ; these modules have this simple form without any other assumptions other than that they must be finitely generated over . 2. 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