Research Article
Finitely Generated Modules over Group Rings of
a Direct Product of Two Cyclic Groups
Ahmed Najim and Mohammed Elhassani Charkani
Department of Mathematics and Informatics, Faculty of Science Dhar Mahraz, Sidi Mohamed Ben Abdellah University,
30000 Fez, Morocco
Correspondence should be addressed to Ahmed Najim; najim.sefrou@gmail.com
Received 27 August 2014; Accepted 15 November 2014; Published 1 December 2014
Academic Editor: Zhongshan Li
Copyright © 2014 A. Najim and M. E. Charkani. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be a commutative feld of characteristic >0 and let =
1
×
2
, where
1
and
2
are two fnite cyclic groups. We give some
structure results of fnitely 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 feld of characteristic >0 and let be a fnite
group. Te study of []-modules in the case where the order
of is divisible by is a very difcult task. When is a
fnite abelian -group, we fnd in [1] the following statement:
a complete classifcation of fnitely 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 fnd this classifcation
in these two cases. Still more, in the case where the Sylow -
subgroup 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]). Tese 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 feld of characteristic >0
and let =
1
×
2
, where
1
and
2
are two fnite cyclic
groups. Let be a fnitely 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. Te 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 satisfed. 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 fnitely generated
over [].
2. Free [
×
]-Modules of Finite Rank
Troughout 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.
Lemma 1 (Nakayama). Let be a -group with p odd,
a ring of characteristic
where is a natural number,
Hindawi Publishing Corporation
Algebra
Volume 2014, Article ID 256020, 5 pages
http://dx.doi.org/10.1155/2014/256020