UNS Conference Portal, IndoMS International Conference on Mathematics and Its Application (IICMA 2021)

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Prime Ideal And Weakly Prime Ideal For Endomorphism Ring (End_Z (M_n (R)),+,∘)
Zakaria Bani Ikhtiyar, Titi Udjiani, Nikken Prima Puspita

Last modified: 2021-11-19


Let R be a commutative ring with the identity. A proper ideal I is called a prime ideal if whenever a,b∈R with ab∈I, then a∈I or b∈I [1,2,3,4,5]. The generalization of a prime ideal is a weakly prime ideal that is restricted to the condition when ab∈I\{0_R} [6]. Given the ring of Z and the set of n×n matrices over the real number, denoted by M_n (R). Since (Z,+,∙) is a ring and (M_n (R),+) is a commutative group, we have M_n (R) is a Z-module [7]. From Z-module M_n (R) we can construct the set of homomorphism from M_n (R) to M_n (R), its called as the endomorphism of M_n (R)  [2, 4]. The set of the endomorphism of  Z-module of M_n (R)  denoted by End_Z (M_n (R))={f|f is a homomorphism from M_n (R) to M_n (R)}, and (End_Z (M_n (R)),+,∘)  is a commutative ring with identity [4]. Ideals from the endomorphism rings has described by Goldsmith and Pabst [8]. We will discussed some properties and characteristics of prime ideal and weakly prime ideal in (End_Z (M_n (R)),+,∘). Here, the ideal P is prime in End_Z (M_n (R))  if and only if End_Z (M_n (R))/P is an integral domain. Moreover, P is a weakly prime ideal of End_Z (M_n (R))  and P is not a prime ideal, then P^2=0_(End_Z (M_n (R)))