Some Properties of Prime Submodules on Dedekind Module
Over Itself
I Gede Adhitya Wisnu Wardhana, Ni Wayan Switrayni and Qurratul Aini
Department of Mathematics, Universitas Mataram Jl.Majapahit 62 Mataram, Indonesia
Keywords: Dedekind Domain, Dedekind Module, Prime Submodules, Principal Ideal Domain.
Abstract: Wardhana et al. gave the characterization of prime submodule and almost prime submodule of finitely
generated module over principal ideal domain in 2016. In this article, we gave some properties of prime
submodule of Dedekind module
over itself.
1 INTRODUCTION
Prime number is the key of Cryptography. Daun gave
the definition of prime submodules in 1978 (Dauns,
1994) and later Khashan gave generalization of prime
submodules in 2012 that called almost prime
submodules (Khashan, 2012). Wardhana et al. gave
characterization of prime submodules and almost
prime submodules in finitely generated module over
principal ideal domain (Wardhana et al., 2018). In
our article we give some properties of prime
submodules in more general case, which is in finitely
generated module over Dedekind domain.
Specifically, we give some properties of prime
submodules in module
over itself.
The definition of Dedekind domain is given
below.
Definition 1 An integral domain is called a
Dedekind domain if satisfying the following three
conditions:
a) Integral domain is a Noetherian ring;
b) Integral domain is integrally closed;
c) Every nonzero prime ideal of is maximal
An example of Dedekind domain is integral
domain
. But
is not Dedekind domain since
is not integrally
closed.
Definition 2 A fraction submodule of -
module , is a set
and denoted by
. A submodule is a prime submodule of
if for every and for all such that
implies or .
For an example, in -module
. Submodule
is a prime submodule of
if and only if is prime.
Submodule
is prime submodule by the definition.
There are lot of properties of Dedekind domain that is
very important. Especially next Theorem.
Theorem 1 Let be a Dedekind domain, if
is ideal of then must be generated by two
elements.
We can see the proof of this Theorem in Dauns
(1994). By Theorem 1, every principal ideal domain
(PID) is Dedekind domain since every ideal of PID is
generated by one element, thus any principal ideal
domain must be generated by two elements by choose
as the second generator. In this article, we used
Dedekind domain
.
Let
is an -module over itself. It is easy
to check that is free module with } as basis.
Corollary 1 A Principal Ideal Domain is a
Dedekind domain.
If is submodule of , then is not always a
free submodule. Furthermore, is not always
generated by one element. For example, let
. Generator of is linearly dependent
since (
. But
is
minimal spanning set of since
and
. This example shows us that
even though the module is free and generated by
290
Wardhana, I., Switrayni, N. and Aini, Q.
Some Properties of Prime Submodules on Dedekind Module Over Itself.
DOI: 10.5220/0008520902900292
In Proceedings of the International Conference on Mathematics and Islam (ICMIs 2018), pages 290-292
ISBN: 978-989-758-407-7
Copyright
c
2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
one element, we can find a submodule of that is not
free and generated by two elements.
Another important property of Dedekind domain
is every nonzero ideal must be product of finitely
prime ideal.
Theorem 2 If is a Dedekind domain, then every
nonzero ideal of is product of finitely many prime
ideals.
In this paper, shall always denote the Dedekind
Domain
and is -module
.
2 RESULT
Given -module , we know that submodule of
are not always generated by one element. But can
be generated by maximum two element.
Theorem 3 Every submodule of an -module
must be generated by at most two elements.
Proof Let be generated by
, .
Let
. Then
. The equality of two complex
number gives us a system linier equation (SLE) with
two equation and variable. According to basic
linear algebra, our SLE always consistent with many
solutions. Hence there is
, , such
that
linear combination of other
. Therefore, we can eliminate
from
, and
is still
generated . If then we can use this
method to eliminate other
until the generator of
is consist by two elements
Now we will give some properties of submodule
that generated by one element. First, we need to
know the fraction of submodule .
Theorem 4 Let be a submodule of -module
. If submodule is generated by
, then
.
Proof Let
and . Suppose
that
for some . Hence,
. Since is arbitrary, then we
have
.
Conversely, let
Choose ,
then . Hence
for some .
Therefore
. So, we have
.
By Theorem 4, it is easy to find fraction
submodule of any submodule that generated by one
element. Theorem 2 also essential to recognize prime
submodule in .
Theorem 5 Let be a submodule of -module
that generated by
. Submodule is
prime if and only if
is prime element in
.
Proof Let
is prime element of . Let
and such that
. We have
, hence . Since is prime, we have
or . According to Theorem 3 we have
or . Hence is prime submodule of
.
Conversely, let be a prime submodule of that
generated by one element. Hence, we can write
. Let and such that . Then
we have for some . Since is prime
then
or
. Therefore
or , and hence is prime element.
In general, submodule need not be generated by
one element. In more general case we have the
following properties of .
Theorem 6 Let be a submodule of -module
. Then
.
Proof Let . Since then we have
. Therefore
. Conversely, let
. Then for any we have , since
is also ideal of . Hence .
Theorem 6 is more general properties of Theorem
4. Hence if is generated by two elements, say
, then we have
.
ACKNOWLEDGEMENTS
This paper is supported by PNBP Universitas
Mataram 2018.
Some Properties of Prime Submodules on Dedekind Module Over Itself
291
REFERENCES
Dauns, J., 1994. Modules and Rings. Cambridge University
Press, New York.
Khashan, Hani A., 2012. On Almost Prime Submodules.
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Wardhana, I G. A. W., Switrayni, N. W., Aini, Q., 2018.
Submodul Prima, Submodul Prima Lemah dan
Submodul Hampir Prima pada Z-modul
.
Eigen Mathematics Journal, 1(1), 2018.
Roman, S., 2007. Advance Linier Algebra, in: Graduated
Text in Mathematics vol. 135. Springer, New York.
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