Al-Jabar: Jurnal Pendidikan Matematika Volume 13, Number 2, 2022, Pages 355 – 361 http://ejournal.radenintan.ac.id/index.php/al-jabar/index 355 Prime ideal on the endomorphism ring of  ℤ (ℤ  ) Zakaria Bani Ikhtiyar 1 *, Nikken Prima Puspita 1 , Titi Udjiani 1 1 Departement of Mathematics, Universitas Diponegoro, Semarang, Indonesia. zakaria.bani45@gmail.com * Abstract Article Information Submitted July 26, 2022 Revised Nov 24, 2022 Accepted Nov 29, 2022 Keywords Endomorphism; Endomorphism Ring; Ideal; Module; Prime Ideal. The set of all endomorphisms over -module  is a non-empty set denoted by   (). From   (), we can construct the ring of   () over addition and composition function. The prime ideal is an ideal that satisfies the properties like prime numbers. In this paper, we take the ring of integer number ℤ and the module of ℤ  over ℤ such that the  ℤ (ℤ  ) is a ring. Furthermore, we show the existence of the prime ideal on the  ℤ (ℤ  ). We also applied a prime ideal property to prime ideal on  ℤ (ℤ  ). INTRODUCTION A module over a ring is a generalization structure of a vector space over a field (Matlis, 1968; Nobusawa, 1964; Wahyuni et al., 2016). A vector space requires a commutative group and a field. In module theory, a field can be replaced with any ring with unity (Marks, 2002; Volodin, 1971; Jensen & Lenzing, 1989). Furthermore, the concept of a linear transformation is known in a vector space. This concept is also implemented in modules called module homomorphism. Let  and  be modules over a ring . If :  → is a module homomorphism in which =, then we call f a module endomorphism (Nicholson, 1976). Here we collect all of the endomorphisms in a module  over ring  such that we have the set of endomorphisms  over  denoted by   () (Lindo, 2017). The endomorphism set is non-empty because at least there is the identity function as an element of   (). Therefore, based on ring theory in Herstein (1975), by adding two binary operations, i.e., addition and composition function, we constructed a ring called an endomorphism ring, denoted by (  (), +,∘). In a ring , a non-empty set ⊆ that satisfies the axioms of the ring and ,  ∈  for all ∈ and ∈ is called ideal (Davvas, 2006; Jianming & Xueling, 2004). Furthermore, the special ideal was defined by Dedekind in 1871 based on the properties of prime numbers. The concept of the prime ideal is known (Kleiner, 1998). Moreover, the prime ideal was discussed by Khariani et al. (2014), Maulana et al. (2019), and Khairunnisa and Wardhana (2019). In this paper, we have a special ring, i.e., the endomorphism ring (  (), +,∘) where =ℤ  and =ℤ. This research shows the existence of a prime ideal on  ℤ (ℤ  ). We also applied a characteristic of prime ideal for the prime ideal on  ℤ (ℤ  ). Here we start our discussion about the prime ideal by giving the fundamental theories of a module over a ring. Definition 1. (Wahyuni et al., 2016) Let (, +,∙) be a ring with unity, (, +) a commutative group, and the scalar operation ⋆ :  ×  →. A group  is a left -module if satisfy the following conditions: i.  1 ⋆ ( 1 + 2 )= 1 ⋆ 1 + 1 ⋆ 2 , How to cite Ikhtiyar, Z, B., Puspita, N, P., & Udjiani, T. (2022). Prime ideal on the endomorphism ring of  ℤ (ℤ  ). Al- Jabar: Jurnal Pendidikan Matematika, 13(2), 355 – 361. e-ISSN 2540-7562. Published by Mathematics Education Department, UIN Raden Intan Lampung.