Completely Prime Modules for Graded Simple Modules Which is Simple Over Leavitt Path Algebra

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ISSN: 2085-8019 (p) ISSN: 2580-278X (e) pp : 1-5

Sainstek: Jurnal Sains dan Teknologi 1

Vol 15 No 1, June 2023 ISSN: 2085-8019 (p), ISSN: 2580-278x (e)

Completely Prime Modules for Graded Simple Modules Which is Simple Over Leavitt Path Algebra

Risnawita

¹

*, Zubaidah

²

*1Prodi Pendidikan Matematika, Fakultas Tarbiyah dan Ilmu Keguruan, UIN Sjech M. Djamil. Djambek Bukittinggi, Sumatera Barat, Indonesia

.

2Prodi Pendidikan Bahasa Arab, Fakultas Tarbiyah dan Ilmu Keguruan, UIN Sjech M. Djamil. Djambek Bukittinggi, Sumatera Barat, Indonesia

.

*email: [email protected]

Article History Received: 8 May 2023 Reviewed: 26 May 2023 Accepted: 14 June 2023 Published: 14 June 2023

Key Words

Completely prime module;

Graded simple module;

Leavitt path Algebra.

Abstract

We characterize a graded simple module in the setting Leavitt path algebras L which is the completely prime module. Let m M and r A, an A-module M is called a completely prime module if for one m M and r A then 0 or m = 0. In this article, we show that If u is a sink or an infinite emitter, then a graded simple module which is simple is not a completely prime module.

INTRODUCTION

The Leavitt path algebra is a branch of representation theory. The Leavitt path algebra was introduced by Abrams and Aranda Pino in 2005 (Abrams & Aranda Pino, 2005), and has become an active area of research recently. Leavitt path algebra is a path algebra on an extended graph that satisfies a certain relation. Let is a Quiver, the path algebra is an algebra over a field with basis is the set of all paths in . A quiver are quadruple of the sets which elements are called vertices, the set which elements are called arrows, and two maps r , which associate to each arrow , its source , and its range , respectively.

Many studies have been carried out on Leavitt path algebra

(Abrams et al., 2015;

Abrams & Aranda Pino, 2005; Ara &

Pardo, 2017; Ara & Rangaswamy, 2014;

Arnone & Cortiñas, 2022; Clark et al., 2016; Hay et al., 2014; Rangaswamy,

2015)

. Abrams, et al in 2007 have studied the necessary and sufficient conditions for a quiver so that the Leavitt path algebra is a finite-

dimensional algebra. Ranggaswamy

(Rangaswamy, 2015) studied the prime ideal characterization of the Leavitt path algebra of any graph. Ara and Rangaswamy

(Ara &

Rangaswamy, 2014)

have also proved that a simple module on Leavitt path algebra is a finitely presented module.

(Abrams & Aranda Pino, 2005)

The prime module consept was

constructed from the structure of prime ideal of a ring [4]. Let be a left module over the ring (written - module ). A proper prime submodule of if with , and implies or (Risnawita et al., 2021). Wardhana (Wardhana et al., 2021) and Saleh (Saleh et al., 2016) were characterized as prime submodules. A module (Risnawita et al., 2021 )is said to be a prime module over ring if with and implies that . The purpose of this study was to

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Sainstek: Jurnal Sains dan Teknologi 2 characterize graded simple modules over

Leavitt path algebra.

In this paper we we will characterize graded simple modules which is simple for a Leavitt path algebras by using a quiver with a sink or infinite emitter. Let is a quiver, A vertex in quiver that does not emit an arrow, we call a sink. A vertex that shoots infinitely many arrows is called an infinite emitter.

In Section 2 of this paper, we will give the method that we use in this research. In section 3, we will give the result and discussion. In Section 4, we give a conclusion about graded simple modules which is simple for a Leavitt path algebras.

METHOD

The research method carried out is in the form of utilizing and developing the results and methods that already exist in the literature, especially the methods used in (Abrams et al., 2015; Chen, 2015; Irawati, 2011; Rangaswamy, 2015; Risnawita et al., 2021; Wardhana et al., 2016) to characterize graded simple modules which is simple for a Leavitt path algebras.

RESULT AND DISCUSSION

A quiver consists of two sets and two maps and . The study The Quiver was studied in papers (Abrams et al., 2011; Pino et al., 2010). If is finite and is finite then is a finite quiver . If is finite for every , then is said to be a row finite quiver. A vertex that does not emit an arrow, we call a sink. A vertex that shoots infinitely many arrows is called an infinite emitter.

In quiver , a path is ordered arrows , with for all . A finite path in quiver is an ordered arrows where for all . In this case, the length of path is , denoted by . Two infinite paths and are tail equivalent, denoted by if , for some integers . A path algebra is well-known in representation theory (Pino et al., 2010). A

finite quiver can be represented as a module.

A representation of can be defined: each vertex was associated with a -vector space and each arrow in associated to a -linear map . Because of a quiver, can be seen as a directed graph, so we can add the edges in opposite direction . The arrows in denoted as the real edge, and then the edges on the opposite are called the ghost edges. The set of all ghost edges in is denoted as ( )^∗.

Following (Chen, 2015) we give a definition of leavitt path algebra of quiver , let k be coefficients as the k-algebra generated by a set { } and { ^∗ which satisfies the following

relations:

(1) s(e)e = e = e r(e), for all e (2) r(e)e^∗ = e^∗ = e^∗s(e), for all e^∗ (3) (CK1) e^∗f = , for all

(4) (CK2) ∑_{ _| ^∗,whenever is not a sink

We denoted this algebra by

.

Moreover, the relation (3) and (4) are called the Cuntz Krieger relations. For further, the Leavitt path algebra

will be written

by

. Every element of can be written as ∑

*, where

, ,

and

are paths in .

Let M is a left , for each , we will define the , with . Through , Chen has introduced some classes of simple modules over Leavitt path algebras (Chen, 2015). By using the Chens method, we will construct simple modules over by using sinks, infinite emitter, or cycles in the quiver

(Ara

& Rangaswamy, 2014; Rangaswamy, 2015, 2020)

Definition 3.1. Let is a sink or an infinite emitter in a quiver . Let be the space having as basis the set { . We construct a left as follows: for each vertex , we define linear transformations and each edge in , linear transformations , and ^∗

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on by defining their actions on basis as follows:

(1) For all ,

(2) { (3) {

(4) ^∗

(5) ∗ {

So it can be easily checked that the endomorphisms { ^∗ ^∗ fulfill relations (1) - (4) of the Leavitt path algebra . This induces an algebra homomorphism from to , mapping to , to and ^∗ to ^∗. Then we can construct as a left module over through the homomorphism .

Furthermore, the following lemma explains that if u is an infinite emitter or a sink then is a simple module.

Lemma 3.2. Let either a sink or an infinite emitter , if u is those vertices then is a simple left .

In (Hazrat, 2013) simple modules have been constructed graded simple modules over L by using the -graded algebraic branching system. The graded simple modules that have been constructed is a graded simple module that is also simple.

Definition 3.3. Let be an arbitrary quiver.

The -algebraic branching system consists of a set X and a family of its subsets { , , , e } such that

(1) , for , with and with (2) for

(3) For all , ⋃ , for all ;

(4) For each e , there exists a bijection

We assume that is saturated, that is,

⋃

Next, given the definition of algebraic branching system.

Definition 3.4. If there is a map then The -branching system is a graded such that

for all e.

Let -algebraic branching system X, we can make a left module over L as follows:

Let K-vector space have X as a basis.

Following (Hazrat, 2013), we can give a definition, for each vertex v linear

and each edge e in ,

transformations linear

and ^∗ n define as follows

:

transformations

For all paths

(1)

{

(2)

{

(3)

^∗ {

It is easy to check that the { ^∗ endomorphisms

fulfill relations (1)–(4) of the Leavitt path algebra L (see

(Hazrat, 2013)

. This induces an algebra homomorphismϕ from L to mapping v to , e to and ^∗ to ^∗. Then we can construct as a left module over L through the homomorphism ϕ. We denote this L-module operation on by ⋅ .

Lemma 3.5. If a sink or an infinite emitter, then module is a graded simple module which is simple.

Next, we will characterize graded simple modules which is simple for a Leavitt path algebras by using a quiver with a sink or infinite emitter.

By using Definition 3.1 and Lemma 3.5, we obtain the following theorem.

Theorem 3.6. If v is a sink, then a graded simple module which is simple is not a completely prime module.

Proof. We can see Figure 1. Let be the quiver in Figure 1.

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Figure 1. Quiver with sink

The module which is simple is the module over , with as basis the set { and is a

sink, actually only contains

{ }. Let ^∗, and , . Then . However, if we take

then So Therefore and , so . Thus, is not a completely module.

So tha the graded simple modules which is simple is not a completely module.

Futhermore, we can check in the case for with subgraph in Figure 2:

Figure 2. Subquiver with sink By using Definition 3.1 and Lemma 3.5, we also obtain the following theorem.

Theorem 3.7. If is an infinite emitter in , then graded simple modules which is simple is not a completely prime module.

Proof. Let see the quiver in Figure 3.

Figure 3. Quiver with infinite emitter The simple module is the module over L, let the basis { . Let and , , thus . However, if we take

then . So . Therefore and such that . Thus, is not a completely module.

CONCLUSION

This paper studied graded simple modules which are simple over Leavitt path algebra. we prove that If u is a sink or infinite emitter, then a graded simple module which is simple is not a completely prime module.

ACKNOWLEDGEMENT

We would like to thank UIN Sjech M.

Djamil Djambek Bukittinggi who supported this research.

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