1. Introduction
Suppose that
the function
is said to be admissible if
and if
then
Admissible functions have been used as a significant tool to determine the structure of abelian p-groups which have certain types of j-diagram series [
1,
2,
3]. Moreover, j-diagrams are used in classifying chain rings and in determining their groups of automorphisms [
4]. Motivated by the important role of j-diagrams in group and ring theory, this article is aimed to investigate the existence and uniqueness of such j-diagrams. We focus our attention on j-diagrams for finite abelian p-groups, and particularly groups of units of finite commutative chain rings. Chain rings are associative rings that have a lattice of ideals that creates a unique chain. A finite ring
R can easily be shown to be a chain ring if and only if its (Jacobson) radical
is principal and
is a field of order
p is prime. Every finite chain ring has five positive integers
named the
invariants. These rings occur in several applications, for details see [
1,
5,
6,
7,
8,
9,
10,
11,
12]. For instance, they have widely appeared in coding theory [
13,
14,
15,
16,
17]. However, the class of Galois rings is a distinguished class of finite chain rings, and every Galois ring is represented as:
where
is a monic irreducible polynomial of degree
Finite chain rings are constructed in at least two different ways. Suppose that
R is a finite chain ring that has the invariants
First,
R can be viewed as an Eisenstein extension of
where
is an Eisenstein polynomial over
i.e.,
where
is a unit of
. Another way to construct
R involves
the field of p-adic numbers. Every chain ring
R is a quotient ring of the integers ring of a certain finite extension of
for more details see [
6] and the references therein. The symmetry of invariants of various chain rings injects more choice and flexibility into the theory of ring construction.
The group of units (multiplicative group)
of
R is defined by
, i.e., the set of all non-nilpotent elements of
From Ayoub (1972) [
2],
where
is cyclic of order
and
which is a p-group. Thus, the structure problem of
is reduced to that of
After Ayoub, we call
H the
one group. If
does not divide
the structure of
H is given by Ayoub [
2] based on the results in [
3]. However, the case when
the full structure of
H is given by Alabiad and Alkhamees [
1]. In this paper, we aim to study the existence and uniqueness of j-diagrams for the one group
H of
In
Section 2, we introduce the concept of j-diagrams, some notations and examples. In
Section 3, we study the existence and uniqueness of complete and incomplete j-diagrams for the series
of the one group
for any finite commutative chain ring
R with invariants
where
Moreover, among other results, we find an explanation of j-diagrams from a ring-theoretic point of view, see Theorem 5.
3. The J-Diagrams for One Group H
In what follows,
R is a finite commutative chain ring with invariants
We focus on the following series of
(
),
where
Definition 3. We call R a complete (incomplete) chain ring if H has complete (incomplete at ) diagram, where .
Lemma 1. If x is a unit in Then, mod if and only if mod
The following lemma is easy to prove.
Lemma 2. Let
- (1)
The map defined by:is an isomorphism.
- (2)
Let defined by: Then, is an isomorphism.
Remark 1. Note that is an elementary p-group.
Theorem 1. Let R be complete, and let for some
- (i)
If then
- (ii)
If and then
- (iii)
If then
Proof. Consider Then, Moreover, with Suppose then As we get Hence, and this ends (i). For (ii), assume that and Then, Now So However, If then gives Similarly, one can prove (iii). □
Lemma 3. Let R be complete.
- (i)
Let e be the smallest positive integer such that Then either or
- (ii)
Any homomorphic image of R is complete.
Proof. (i)
As
by definition
For any
for some
with
Consider any
then
If
then
Suppose that
Then,
gives
(ii) Let
I be a non-zero ideal of
for some
For
and
for
Thus, whenever
and
For some
suppose that
Then,
However, either
or
under the mapping
Hence,
This shows that
and thus
T is complete with the admissible function
defined on
as follows:
if
otherwise
□
Lemma 4. Let R be complete, and let i.e., If then the followings hold
- (i)
- (ii)
- (iii)
There exists a unit such that and where
Conversely, if there exists with and if R satisfies (i), (ii) and (iii), then there exists an admissible function j on for which R is complete.
Proof. Now
If
by Theorem 1,
As
we get
This is a contradiction. Hence,
and
For any
the binomial expansion gives
for some
with
Consider any unit
As
then,
So,
and in particular
It follows that
and hence
This proves
The hypothesis gives that
for some unit
Then,
gives that
For the converse, define
j such that
for
and
for
Then for any unit
As
Hence,
By using this, and the argument in [
2], it can be easily verified that
j is an admissible function and that
R is complete. □
Theorem 2. Let R be a finite commutative chain ring with invariants and for some Then there exists an admissible function j on such that and R is complete if and only if R satisfies the following conditions:
- (i)
- (ii)
- (iii)
in
Proof. Let
R be complete and
Let
such that
By Lemma 4, (i) and (ii) hold, and there exists a unit
such that
and
Now
for some unit
Then,
It follows that
In
by (ii),
so
, i.e.,
Conversely, let
R satisfy (i), (ii) and (iii). By (iii),
Hence,
for some
Consider
Then,
and
By Lemma 4, the desired
j exists. □
Theorem 3. Let R be a finite commutative chain ring with invariants and for some Then there exists an admissible function j on such that and R is complete if and only if
Proof. Suppose that
R is complete and
For any
for some
with
Fix an
As
then we get
Moreover,
for some unit
So
and thus
is a unit. For any
as
has weight
so
is a unit. Thus, in
Now,
for some unit
Then,
and
so
Thus,
As
we get
Consequently,
Conversely, let
Consider
then for any unit
is a unit. It follows that for
has weight
and thus
For
define
and for
define
. By using ([
2], Propositions 1 and 2), it follows that
j is the desired admissible function. □
Theorem 4. Let R be a finite commutative chain ring with invariants . If R is complete, then there exists only one admissible function j on
Proof. Suppose that
Then,
Using this it follows that any finite chain ring
R of characteristic
p is complete and the underlying admissible function
j on
is such that
whenever
and
otherwise. Suppose
and
are two different admissible functions on
such that
R is complete with respect to
j as well as
It follows from the proof of Theorem 1 that if for some
and
then
If for some
and
then
So, there exists an
such that
and
It follows from Lemma 4 that
and we can take
Let
and
be the restriction of
j and
, respectively, to
Set
By applying (Theorem 1 (4) [
3]), we get two sets of cyclic
subgroups
of
H corresponding to
j and
, respectively. By Proposition 6,
so
Furthermore,
gives
This is a contradiction, and thus proves the result. □
Remark 2. By a similar discussion, the above results hold if we assume that R is incomplete.
Remark 3. Consider a finite commutative chain ring R with . Let be an Eisenstein polynomial of By looking at the invariants and the element one knows whether a given R is complete or incomplete using Theorems 2 and 3. In any case, the form of the underlying admissible function j on is well defined by:where where Example 4. Let R be a chain ring with invariants and suppose that and Then, R is clearly a complete j-diagram with unique admissible function j. This means if there is another admissible function j’ such that R is also a complete j’-diagram, then For the converse, note that if which is an admissible function but R is not diagram. This means the existence of an admissible function j on is not enough to say R is j-diagram (either complete or not), see Definition 2. Thus, in general, the converse is not true.
Proposition 1. Let R be a finite commutative chain ring with If or then R is complete.
Proof. Note that for
where
Thus, if
then clearly the series (
7) is a complete j-diagram, and hence
R is complete. Now, assume that
If
,
and hence
It follows from Equation (
11) that
for some
Furthermore, when
where
and
are defined in Lemma 2 Thus,
is an isomorphism. In case of
consider the map
One can prove easily that is well-defined, and moreover, is a monomorphism. For epimorphism, note that since is a finite field, then a basic field. That is, if then mod where . Then, there is such that and then mod Therefore, is an isomorphism and which means is an isomorphism. □
Corollary 1. Any finite commutative chain ring R with characteristic p is complete.
Proof. Since then which means that and by Proposition 1, R is complete. □
Remark 4. By Proposition 1, when the j-diagram for H is independent of the Eisenstein polynomial However, this is not true when , i.e., Let by Equation (11),Thus, mod if and only if mod , i.e., in Proposition 2. If then is an isomorphism if and only if
Proof. If is an isomorphism, then ker which means that mod for any Hence, has no zeros in and thus The converse is direct by Theorem 3. □
The following theorem gives a characterization of incomplete chain rings.
Theorem 5. Suppose that R has invaraints with Therefore, the subsequent hypotheses are equivalent:
- (i)
R is incomplete.
- (ii)
There is such that
- (iii)
divides k and there exists such that
Proof. Let (i) be satisfied, thus ker
because
is surjective. In this case, there is
in ker
with
where
However, the above equation holds when
and
mod
Now, assume that
then ker
ker
where
f is a homomorphism;
and
Moreover, ker
if and only if
has only zero solution. Thus,
is a root of
in
where
and
The remaining hypotheses follow immediately by Proposition 2 and Theorem 3. □
Corollary 2. If R is an incomplete chain ring, then ker is of rank
Proof. Since any element in ker is of the form where is a zero of the polynomial Thus, the order of ker is exactly p since there are p distinct zeros of in □
Lemma 5. Let R be a finite commutative chain ring with invaraints
- (a)
If or and Then,
- (b)
If Then,
Proof. Part (b) is obvious; note that if
then
. For part (a),
However, and then thus, □
Proposition 3. If then Furthermore,where means the greatest integer that is less than or equal to Proof. Let
for some
then clearly
which means
If
then it is clear that
Thus,
For the case
and thus,
□
Proposition 4. Assume the admissible function j satisfies: if then for all Then, in particular,
Proof. The proof is conducted by induction on
First, let
and note that
If
then
where
and
Moreover,
for some
and
where
and
Since
it follows that
are elements of
This means
As we proceed, we get
and thus
Therefore,
If
observe that
and hence the conclusion is drawn from the induction step. □
Next, we give an important result; that is useful in capturing the structure of the subgroups
of
H via the following j-subdiagram:
Which in turn helps us to investigate the group of automorphisms of
for more details see Remark 4.2.10 in [
4].
The following result for finite abelian groups can be easily proved.
Lemma 6. Let G be a finite direct product of cyclic groups, each of order for some Let be a subgroup of G which is a direct product of cyclic groups each of order and for which Then, such that each is a cyclic group and Moreover, for any where
Theorem 6. Let be a complete diagram for an abelian p-group A, and Let and Then,
- (a)
- (b)
satisfying the following conditions:
- (i)
- (ii)
There exists a subset D of disjoint form B and a one to one mapping such that for any for some Suppose and
- (iii)
and
- (iv)
Let and Then,
Proof. (a)
Put
Then,
is a complete
diagram, where
is given by
Note that
By [Theorem 1 [
3]],
(b) Write
with
and
It is clear that
Suppose that all
have the same rank. For each
there exists a positive integer
and a unique
such that
Thus,
It follows that from the definition of
is disjoint from
B and there exists a bijection
such that
This proves (ii). Hence,
This proves (iii). Finally, consider
and
Observe that any
is in
G if and only if each of its components in the decomposition
is in
For any
implies
For any
whenever
So
Consider any
Now,
As the order of
is
and the order of
is
then by Lemma 6,
□