1. Introduction
In this article, we consider finite commuative chain rings, although some results still correct under more general situation. Ayoub called these rings (cf. [
1]) primary homogeneous rings. Such rings arise in various places; in Coding Theory (cf. for example [
2,
3]); in Geometry (cf. [
4]). It is well-known that every finite chain ring
R is an Eisenstein extension of some degree
k over a Galois ring of the form
There are positive integers
and
m associated with
R, called invariants of
Our main aim in this study is to obtain the structure of the multiplicative group
of
In (1972) (cf. [
1]), Ayoub obtained various results based on ideas (cf. [
5]) regarding j-diagrams for abelian
p-groups.
One major result claims that the factorization of an abelian
group with incomplete j-diagram can be completely obtained by the mentioned diagram. This idea was then used to find the structure of
when
R is not necessarily finite (Theorem 3,
Section 4 [
1]). However, later, it turned out that incomplete j-diagrams failed to determine the exact decomposition of multiplicative groups of some examples introduced by Luis [
6]. On the other hand, Hou [
7] by different approach gave the structure of
in a special case;
This sudy demonstrates that the result (Theorem 3, [
1]) is still valid in some cases, for instance the case studied in [
7]. Additionally, we enhance the idea that incomplete j-diagrams are generally not enough to obtain the decomposition of bounded
p-groups. That is, there is a relation amongst the generators which plays essential role in determining the structure. That relation depends not only on the related incomplete j-diagram, but also on the Eisenstein polynomial by which
R is constructed over its coefficient subring
However, with this relation being taken into consideration, we manage to make incomplete j-diagrams succeed in recapturing the structure of multiplcative groups of finite chain rings. In addition, a set of linearly independent generators for
is provided and, further, the number of such generators is computed for each possible order. Finally, we give the correct version of ([
5], p. 458).
2. Preliminaries and Notations
In what follows,
R will denote a finite commutative chain ring of characteristic
and nonzero radical
N (the case when
, R is a field) with index of nilpotency
The residue field
is a finite field of order
We now state some facts and introduce notations that we shall use throughout. For the details, we refer the reader to [
8] for finite chain rings and [
1,
5] for j-diagrams.
The ring
R contains a subring (coefficient subring)
S of the form
where
a is an element of
S of multiplicative order
If
it is easy to see that
, since
is of 1 dimension over
In this case,
R is expressed as:
(as
module) where
k is the greatest integer
such that
It follows that
where
This means that
is a root of an Eisenstein polynomial over
S of the form:
The positive integers
and
m are called invariants of
Furthermore,
for some
Let
and consider the following filtering:
joined with a function
j defined as:
where
Note that if Char
i.e.,
put
The series (
4) with
j defined in (
5) and the
p-th power homomorphisms
from
into
form what we call
j-diagram. However, we refer to the series (
4) when we mention j-diagram. If there exists
, such that
is not an isomorphism, we then call the series (
4) incomplete
j-diagram at
and complete
j-diagram otherwise (all
are isomorphisms).
We denote, by
, the units group of
R, and so one can verify that
where
is a cyclic group of order
and
is the
p-Sylow subgroup of
The structure problem of
is then reduced to that of
After Ayoub [
1] we call
H the One Group of
If
is the number of basis elements in
H whose orders
then
H is decomposed as:
where
is a cyclic group of order
We define
p-rank
of
H as
If
N
fixed, then from Eisenstein polynomial (
3)
where
and
Now, let
be a representatives system in
R for a basis of
F over its prime field (
). Set
where
is the range of
For each
let
be a subgroup of
H generated by
Hence one can easily show that
Now, if (
4) is a complete
diagram, then, from (cf. [
1]), the system
(*) forms a basis for
Proposition 1 (Theorem 3, [
1])
. Assume that (4) is a complete j-diagram. Then, The purpose of the present paper is to establish a basis for
H when (
4) is an incomplete j-diagram at
Moreover,
and
are determined in both cases (complete and incomplete) in Theorems 3 and 4.
Unless otherwise mentioned, all of the symbols stated above will retain their meanings throughout and, additionally, if is the least positive number fulfilling
3. The Determination of the Structure of
Throughout this section, where and The following proposition is useful and it will be needed later.
Proposition 2. If then the following conditions are equivalent:
(i) The series (4) is an incomplete j-diagram at (ii) The polynomial has a root in
(iii) and
Proof. Assume that (i) holds, then ker since is onto. Then there is ker , such that where This only happens when and mod Hence, if ker is isomorhpic to that of the homomorphism, defined by: However, ker if and only if has nonzero solutions in The remaining hypotheses follow immediately. □
Definition 1. We call R incomplete (complete) chain ring if one of the equivalent conditions in Proposition 2 holds (otherwise).
Corollary 1. If R is an incomplete chain ring, then ker is of rank
Remark 1. In the light of Proposition 2 and its corollary, and i.e., Furthermore, if η is the composition of then ker η is of rank Letbe a generator of ker Assume that
f is as mentioned in the proof of Proposition 2 and
. Construct the following system:
The following proposition is stated without proof, since it involves the same ideas of that in the complete case.
Proposition 3. The set () represents a system of generators for
Observe that
ker
and then
where
and
are positive integers that are divisible by
i.e.,
Note that
will keep its meaning throughout the paper, as described here. Let
be a subgroup of
H that is generated by
Thus, by (cf. [
1]),
has a direct decomposition,
Theorem 1. Let R be an incomplete chain ring and thenwhere D is a cyclic group of order Proof. By Equation (
13), it is clear that
and also
is a p-pure subgroup of
every element is not of a smaller p-height in
than in
Thus, by the results from (cf. [
9]),
is a direct summand of
This finishes the proof. □
Corollary 2. If then is a basis for
Corollary 3. The system is a basis for H if and only if (the order of ξ).
Remark 2. - (1)
It is worthy to mention that Theorem 1 coincides with Ayoub’s results (Theorem 3 Section 4, [
1]
). That means in such case, incomplete j-diagrams succeed in retrieving the structure of - (2)
If then obviously and, thus, the result that is given in [
7]
is just a particular case of Theorem 1.
From now on, we assume
Consider Equation (
13) and let
Lemma 1. If R is an incomplete chain ring and then there is a positive integer d and p-pure subgroups of H, such that
Proof. For
let
where
is the p-adic valuation. Subsequently, it is clear that
for all
Choose
such that
If there are more than one of
s satisfying
choose the smallest one. Let
be defined as
If
and
, then Equation (
13) can be rewritten as:
where
and
Define
to be the subgroup of
H that is generated by
Thus, from (
17),
Because
then the p-height of
in
is not lesser than that in
H and thus
is a p-pure subgroup of
Now, if we raise Equation (
17) to
we obtain a new relation among the generators of
where
and
, because some generators might disappear due to raising (
17) to a power of
Note that since the generators on the right side of Equation (
18) are linearly independent, thus the left side cannot vanish. Again, there is
such that
is the smallest
(
) and then
is the smallest
Assume that
then we have a similar situation to that of
Let
and
be a subgroup of
generated by
and all
with
By a similar arguement,
is a p-pure subgroup of
and, thus, transitively (Lemma 26.1, [
9])
is a p-pure subgroup of
Therefore, after a finite number
d of similar processes, we obtain:
□
Remark 3. Based on notations that were introduced in Lemma 1, put and letwhere and Theorem 2. Let R be an incomplete finite chain ring with invariants such that Then, - (i)
is a direct product of r cyclic groups of order
- (ii)
is a direct product of cyclic groups of order and is a cyclic group of order
- (iii)
D is a cyclic group of order
- (iv)
C is a cyclic group of order if and otherwise.
Proof. Because, for each
is a p-pure subgroup of
(Lemma 1) and also a bounded p-group, then (Theorem 27.5, [
9])
is a direct summand of
there is a subgroup
, such that
Thus, based on (
20) and Remark 3,
By the argument of the proof of Lemma 1,
Hence,
where
is a cyclic group of order
Therefore, the decomposition (
22) is just a result of successive application to the above argument. Note that
□
Corollary 4. With the same assumption in Theorem 2,
Corollary 5. (), ξ and η if form a basis of
The following example demonstrates the failure of Ayoub’s results (Theorem 3, [
1]); non-isomorphic abelian p-groups may have the same incomplete j-diagrams. Furthermore, it shows how Eisenstein polynomials play a key role in factorizing
Example 1. Let R be a finite chain ring with invariants and Forward computation leads towhere Note that here is not p-pure subgroup of Since and thus from Theorem 2 Now, if we consider T is a finite chain ring with same invariants (same incomplete j-diagram) and different Eisenstein polynomial Let where (root of ) and a is a root of Then, and, hence, from Theorem 1, Next, we state the correct version of (cf. p. 458, [
1]).
Corollary 6. Let R and T be two incomplete chain rings with same invariants and same Ω and Then,
In general, Lemma 1 gives an algorithm for calculating and s. The following example illustrates that algorithm.
Example 2. Let R be an incomplete chain ring with invariants and where and is an Eisenstein extension over S by i.e., is a root of (Proposition 2). Clearly, is an incomplete chain ring with where and Consider the (admissible) function with A similar equation of that in (13),where Let be an Eisenstein extension over by where is an incomplete chain ring with invariants , such that Now, put in (26) and after forward computations we getwhere and If then based on Lemma 1 Continuing in this way and after l steps (), R is an Eisenstein extension over by where Thus, as a result where for 4. Enumeration of Generators of the Same Order
In this section, we compute p-rank
and
for the decomposition that is given in (
15) and (
22). However, first we determine
and
for the complete case (
10). If we denote
then using the j-diagram it is not hard to prove the following:
where
means the greatest positive integer that is less than or equal to
Let
Note that, if By the j-diagram, one can easily prove that when and when Thus, for if and otherwise. Hence, it suffices to obtain
Theorem 3. Assume that R is a complete chain ring, then and if Proof. The first claim is easy. Note that
is exactly the cardinality of
For each
s, there is a positive integer
, such
and, hence,
However, since
or
then, if
, either
or
Consider two cases: (i) let
then clearly
for all
and, thus, by the definition of
However, we know that
and, hence,
(ii) Assume
then if
by definition of
Also if
which means that
hence
On the other hand, Thus, in this case From (i) and (ii), we conclude that, for every That implies A is the set of all positive integers in the interval except those in i.e., mod Therefore, the proof follows. □
Corollary 7. Based on the assumptions of Theorem 3, We now compute
for the incomplete case. In this case,
, which means either
and
or
Put
where
Proposition 4. Let and Then with same notations mentioned above, Proof. By the j-diagram, one can check that is a subgroup of Because there is an element in H with (Theorem 2), then clearly and On the other hand, the proof of Lemma 1 implies for all (since ) and, hence, we only consider This immediately implies for all e other than and □
Remark 4. By Proposition 4, when , it suffices to compute However, in case of nothing can be said since
Note that, in the case when
once we determine (
13) and
for the part
we then completely obtain
Denote
where
Theorem 4. Let R be an incomplete chain ring, then
(i)
Proof. (i) is obvious. For (ii), we first let
Note that
and
if
thus we have
Now, consider two cases: (a) when
then
and clearly
if and only if
Consequently, there are
of such
s and, thus
(b) In the case when
Firstly, observe that
if and only if
By the same reasoning of Case (a), we obtain the results. Secondly, when
in this case all
s satisfying
occur in the interval
if
or
when
Therefore, we return to a similar situation of Theorem 3 with
and
instead of
and
e, respectively. Observe that, if
and
then there exist
generators of
and then put
When
we have to add
, since
On the other hand, if
there are
of
each having
elements of basis of the same order, so put
□
In conclusion, this article considered the multiplicative groups of finite commutative chain rings. The case when was completed for every incomplete chain ring. Moreover, a set of linearly independent generators for H was established by connecting those generators to a basis of F over its prime field.
Recently, there has been increasing interest in using finite commutative chain rings in Coding Theory. Researchers may wish to further apply this knowledge about finite commutative chain rings to Coding Theory.