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Journal : Sains dan Matematika

 1 
BENTUK-BENTUK IDEAL PADA SEMIRING (Z+, +,.) DAN SEMIRING (Z+, ⊕, *) Setyawati, Dian Winda; Soleha, Soleha; Rimadhany, Ruzika
Sains & Matematika Vol 3, No 1 (2014): Oktober, Sains & Matematika
Publisher : Sains & Matematika

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Abstract

Himpunan bilangan bulat taknegatif, yaitu (Z+,+,.) merupakan semiring terhadap operasi penjumlahan dan perkalian biasa, sedangkan himpunan (Z+,?,*) juga merupakan semiring terhadap operasi penjumlahan ? dan perkalian yang didefi nisikan sebagai berikut: untuk setiap a,b?Z+ berlaku a ? b=FPB(a,b) dan a*b=KPK(a,b). Pada semiring R, himpunan bagian I dari R disebut ideal pada R jika a,b ? I dan r ? R maka a+b ? I dan ra, ar ? I . Pada artikel ini ditunjukkan bentukbentuk ideal pada semiring (Z+,+,.) dan bentuk-bentuk ideal pada semiring (Z+,?,*) serta menunjukkan hubungan satu ideal dengan ideal yang lain. Bentuk-bentuk ideal yang ditunjukkan adalah ideal maksimal, ideal substraktif, Q-ideal, ideal prima, ideal semiprima dan ideal primary. The set of nonnegative integers (Z+,+,.) is a semiring of the usual operations of addition and multiplication otherwise set (Z+,?,*) is also a semiring of the addition operation ? and multiplication defi ned as follows: for each a,b?Z+applies a ? b=gcd(a,b) and a*b=lcm(a,b). At semiring R, a subset I of Ris called an ideal in R if a,b ? I and r ? R, then a+b ? I and ra, ar ? I In this paper will be shown the forms of the ideal on the semiring (Z+,+,.) and forms of the ideal on the semiring (Z+,?,*) and shows the relationship of the ideal with the other ideal. Ideal form sthat will be shown is the maximal ideal, substractive ideal, Q-ideal, prime ideal, and the semiprime ideal and, primary ideal.
Bentuk-bentuk Ideal pada Semiring (Z+, +,.) dan Semiring (Z+, ⊕, *) Setyawati, Dian Winda; Soleha, Soleha; Rimadhany, Ruzika
Sains & Matematika Vol 3, No 1 (2014): Oktober, Sains & Matematika
Publisher : Sains & Matematika

Show Abstract | Download Original | Original Source | Check in Google Scholar

Abstract

Himpunan bilangan bulat taknegatif, yaitu (Z+,+,.) merupakan semiring terhadap operasi penjumlahan dan perkalian biasa, sedangkan himpunan (Z+,⊕,*) juga merupakan semiring terhadap operasi penjumlahan ⊕ dan perkalian yang didefi nisikan sebagai berikut: untuk setiap a,b∈Z+ berlaku a ⊕ b=FPB(a,b) dan a*b=KPK(a,b). Pada semiring R, himpunan bagian I dari R disebut ideal pada R jika a,b ∈ I dan r ∈ R maka a+b ∈ I dan ra, ar ∈ I . Pada artikel ini ditunjukkan bentukbentuk ideal pada semiring (Z+,+,.) dan bentuk-bentuk ideal pada semiring (Z+,⊕,*) serta menunjukkan hubungan satu ideal dengan ideal yang lain. Bentuk-bentuk ideal yang ditunjukkan adalah ideal maksimal, ideal substraktif, Q-ideal, ideal prima, ideal semiprima dan ideal primary. The set of nonnegative integers (Z+,+,.) is a semiring of the usual operations of addition and multiplication otherwise set (Z+,⊕,*) is also a semiring of the addition operation ⊕ and multiplication defi ned as follows: for each a,b∈Z+applies a ⊕ b=gcd(a,b) and a*b=lcm(a,b). At semiring R, a subset I of Ris called an ideal in R if a,b ∈ I and r ∈ R, then a+b ∈ I and ra, ar ∈ I In this paper will be shown the forms of the ideal on the semiring (Z+,+,.) and forms of the ideal on the semiring (Z+,⊕,*) and shows the relationship of the ideal with the other ideal. Ideal form sthat will be shown is the maximal ideal, substractive ideal, Q-ideal, prime ideal, and the semiprime ideal and, primary ideal.
Co-Authors Adi Maulana Rachman Adian Husaini Adipati Napoleon Adrian Adrian Agus Hermawan Ahmad Zaenudin Aisyah Aisyah Aldy, Fahri Aminah Aminah Ansyori, Ansyori Apolinaris S Awotkay Aprianti, Kaiza Nurul Bakri Bakri Bambang Kurniawan Barokah, Mutiara Chairul Imron, Chairul Deni Wahyudi Dian Winda Setyawati Eka Wahyu Ningsih Elis Chalisyah Endin Mujahidin erfandari, ovy Erika Setia ningsih Evahelda Evahelda Fadhila, Rahma Fahim, Kistosil Fahmi, Putri Mariska Fathurrosi, Fathurrosi Febrino, Febrino Firmansyah Firmansyah FX. Didik Purwosetiyono Gusti Putu Eka Kusuma Habaridota, Maha Lastasa Buju Basafpipana Hadarah, Hadarah Hanafiah, Maulidya Hasriani Hasriani, Hasriani Hatamar, Hatamar Iis Herisman Imanudin, M.S. Indriani, Nimas Kusumajaya, Hendra Lamdo, Herfandi Mar'ah, Dian Latifathul Marlina, Tuti Maryam Maryam Maryana Maryana, Maryana Mar’ah, Dian Latifathul Mufid, Muhammad Syifa'ul Nabillah Anissa Nataris, Galuh Bintang Nona, Nona Nugraha, Jodi Nurhasanah, Iis Ariska Nuryati Nuryati Prihandono, Prihandono Putri, Ami Manda Atika Putu Nita Yulianti, Ni Luh Ranaswijaya, Ranaswijaya Rasiman Rasiman Retno Wulansari Rimadhany, Ruzika Rinurwati Rinurwati Ripana, Wida Saefuddin, Didin Salsabila, Nadhira Ghea Sanjaya, Refki Saputra, Ahmad Rizki Sembiring, Indrila Sibuea, Masrul Zuhri Sineba Arli Silvia, Sineba Arli Sinulingga, Rani Anjani siska dwi paramitha Subiono - Sumartono, Akhmad Syahfitri Purnama Tika Leoni Putri Usadha, I Gst Ngr Rai Utami, Indah Febrianti Widiyani, Dimas Prakoswo Yudi Sapta Pranoto Yunida, Fera