Ideal Prima Pada Daerah Dedekind Berupa Polinom Faktor Berderajat Satu Dari Ring Bilangan Bulat
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Muklas Maulana(1*)
Universitas Mataram
(*) Corresponding Author
Keywords:
Dedekind domain, prime ideal, maximum ideal, integrally closed
Abstract
Dedekind Domain and prime ideals are part of the topics discussed in the field of abstract algebra. An integral Domain is said to be the Dedekind Domain if and only if it is a Noetherian, integrally closed, and each of its prime ideals is a maximum ideal. In 2019, Maulana discussed the prime ideal’s properties of Gauss integers. At present, there is no research about prime ideals of specific Dedekind Domain, because of that reason, in this article we will give some prime ideals and prime ideals’ characteristics in the area of the Dedekind Domain Z[x]/<x^2>. In this article, it is found the conclusion I=<x> and I=<k,x> are prime ideals in that Dedekind Domain.
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References
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Berrick, A.J., & Keating, M.E. (2000). Cambridge Studies in Advanced Mathematics 65 An Introduction to Rings and Modules with K-Theory in View. United Kingdom: Cambridge University Press.
Fraleigh, J.B. (2013). A First Course in Abstract Algebra (7th ed). United Kingdom: Pearson Education Limited.
Gazir, A., & Wardhana, I. G. A. W. (2019). Subgrup Non Trivial Dari Grup Dihedral. Eigen Mathematics Journal 1(2), 73-76.
Herstein, I.N. (1975). Topics in Algebra (2nd ed). United States of America: John Wiley & Sons.
Hijriati, N., Wahyuni, S., & Wijayanti, I. E. (2018, September). Injectivity and Projectivity Properties of the Category of Representation Modules of Rings. In Journal of Physics: Conference Series (Vol. 1097, No. 1, p. 012078). IOP Publishing.
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Maulana, F., Wardhana, I. G. A. W., & Switrayni, N. W. (2019). Ekivalensi Ideal Hampir Prima dan Ideal Prima pada Bilangan Bulat Gauss. Eigen Mathematics Journal, 1-5.
Misuki, W. U., Wardhana, I. G. A. W., & Switrayni, N. W. (2021, March). Some Characteristics of Prime Cyclic Ideal On Gaussian Integer Ring Modulo. In IOP Conference Series: Materials Science and Engineering (Vol. 1115, No. 1, p. 012084). IOP Publishing.
Kleiner, I. (2007). A History of Abstract Algebra. United States of America: Birkh user Boston.
Rotman, J.J. (2005). A First Course in Abstract Algebra (3rd ed). United States of America: Pearson Education Limited.
Saleh, K., Astuti, P., & Muchtadi-Alamsyah, I. (2016). On the structure of finitely generated primary modules. JP Journal of Algebra, Number Theory and Applications, 38(5), 519.
Sivaramakrishnan, R. (2019). Certain Number-Theoretic Episodes in Algebra(2nd ed). United States of America: Chapman and Hall/CRC.
Switrayni, N. W., Wardhana, I. G. A. W., & Aini, Q. (2022). On Cyclic Decomposition Of Z-Module M_ {m×r}(Z_n), Journal of Fundamental Mathematics and Applications, 5(1), 47-51
Wahyuni, S., Wijayanti, I. E., Yuwaningsih, D. A., & Hartanto, A. D. (2021). Teori ring dan modul. UGM PRESS.
Wardhana, I., & Astuti, P. (2015). Karakteristik Submodul Prima Lemah dan Submodul Hampir Prima pada. Jurnal Matematika & Sains, 19(1), 16-20
Wardhana, I. G. A. W., Astuti, P., & Muchtadi-Alamsyah, I. (2016). On almost prime submodules of a module over a principal ideal domain. JP Journal of Algebra, Number Theory and Applications, 38(2), 121-138.
Wardhana, I. G. A. W., Nghiem, N. D. H., Switrayni, N. W., & Aini, Q. (2021, November). A note on almost prime submodule of CSM module over principal ideal domain. In Journal of Physics: Conference Series (Vol.2106, No. 1, p. 012011). IOP Publishing.
Wardhana, I. G. A. W. W., & Maulana, F. (2021). Sebuah Karakteristik dari Modul Uniserial dan Gelanggang Uniserial. Unisda Journal of Mathematics and Computer Science (UJMC), 7(2), 9-18.
Wardhana, I. G. A. W. (2022). The Decomposition of a Finitely Generated Module over Some Special Ring. JTAM (Jurnal Teori dan Aplikasi Matematika), 6(2), 261-267.
Wijayanti, I. E., & Wisbauer, R. (2009). On coprime modules and comodules. Communications in Algebra, 37(4), 1308-1333.
Yuwaningsih, D. A., & Wijayanti, I. E. (2015). On jointly prime radicals of (R, S)-modules. Journal of the Indonesian Mathematical Society, 25-34
Amir, A. K., Astuti, P., & Muchtadi-Alamsyah, I. (2010). Minimal Prime Ideals of Ore Extensions over Commutative Dedekind Domains. arXiv preprint arXiv:1002.0278.
Astuti, P., & Wimmer, H. K. (2006). Regular submodules of torsion modules over a discrete valuation domain. Czechoslovak Mathematical Journal, 56(2), 349-357.
Berrick, A.J., & Keating, M.E. (2000). Cambridge Studies in Advanced Mathematics 65 An Introduction to Rings and Modules with K-Theory in View. United Kingdom: Cambridge University Press.
Fraleigh, J.B. (2013). A First Course in Abstract Algebra (7th ed). United Kingdom: Pearson Education Limited.
Gazir, A., & Wardhana, I. G. A. W. (2019). Subgrup Non Trivial Dari Grup Dihedral. Eigen Mathematics Journal 1(2), 73-76.
Herstein, I.N. (1975). Topics in Algebra (2nd ed). United States of America: John Wiley & Sons.
Hijriati, N., Wahyuni, S., & Wijayanti, I. E. (2018, September). Injectivity and Projectivity Properties of the Category of Representation Modules of Rings. In Journal of Physics: Conference Series (Vol. 1097, No. 1, p. 012078). IOP Publishing.
Juliana, R., Wardhana, I. G. A. W., & Irwansyah. (2021, February). Some characteristics of cyclic prime, weakly prime and almost prime submodule of Gaussian integer modulo over integer. In AIP Conference Proceedings (Vol. 2329, No. 1, p. 020004). AIP Publishing LLC.
Maulana, F., Wardhana, I. G. A. W., Switrayni, N. W., & Aini, Q. (2018). Bilangan Prima dan Bilangan tak Tereduksi pada Bilangan bulat Gauss. In Prosiding Seminar Nasional APPPI II (pp. 383-387).
Maulana, F., Wardhana, I. G. A. W., & Switrayni, N. W. (2019). Ekivalensi Ideal Hampir Prima dan Ideal Prima pada Bilangan Bulat Gauss. Eigen Mathematics Journal, 1-5.
Misuki, W. U., Wardhana, I. G. A. W., & Switrayni, N. W. (2021, March). Some Characteristics of Prime Cyclic Ideal On Gaussian Integer Ring Modulo. In IOP Conference Series: Materials Science and Engineering (Vol. 1115, No. 1, p. 012084). IOP Publishing.
Kleiner, I. (2007). A History of Abstract Algebra. United States of America: Birkh user Boston.
Rotman, J.J. (2005). A First Course in Abstract Algebra (3rd ed). United States of America: Pearson Education Limited.
Saleh, K., Astuti, P., & Muchtadi-Alamsyah, I. (2016). On the structure of finitely generated primary modules. JP Journal of Algebra, Number Theory and Applications, 38(5), 519.
Sivaramakrishnan, R. (2019). Certain Number-Theoretic Episodes in Algebra(2nd ed). United States of America: Chapman and Hall/CRC.
Switrayni, N. W., Wardhana, I. G. A. W., & Aini, Q. (2022). On Cyclic Decomposition Of Z-Module M_ {m×r}(Z_n), Journal of Fundamental Mathematics and Applications, 5(1), 47-51
Wahyuni, S., Wijayanti, I. E., Yuwaningsih, D. A., & Hartanto, A. D. (2021). Teori ring dan modul. UGM PRESS.
Wardhana, I., & Astuti, P. (2015). Karakteristik Submodul Prima Lemah dan Submodul Hampir Prima pada. Jurnal Matematika & Sains, 19(1), 16-20
Wardhana, I. G. A. W., Astuti, P., & Muchtadi-Alamsyah, I. (2016). On almost prime submodules of a module over a principal ideal domain. JP Journal of Algebra, Number Theory and Applications, 38(2), 121-138.
Wardhana, I. G. A. W., Nghiem, N. D. H., Switrayni, N. W., & Aini, Q. (2021, November). A note on almost prime submodule of CSM module over principal ideal domain. In Journal of Physics: Conference Series (Vol.2106, No. 1, p. 012011). IOP Publishing.
Wardhana, I. G. A. W. W., & Maulana, F. (2021). Sebuah Karakteristik dari Modul Uniserial dan Gelanggang Uniserial. Unisda Journal of Mathematics and Computer Science (UJMC), 7(2), 9-18.
Wardhana, I. G. A. W. (2022). The Decomposition of a Finitely Generated Module over Some Special Ring. JTAM (Jurnal Teori dan Aplikasi Matematika), 6(2), 261-267.
Wijayanti, I. E., & Wisbauer, R. (2009). On coprime modules and comodules. Communications in Algebra, 37(4), 1308-1333.
Yuwaningsih, D. A., & Wijayanti, I. E. (2015). On jointly prime radicals of (R, S)-modules. Journal of the Indonesian Mathematical Society, 25-34
Published
2022-12-07
How to Cite
Maulana, M. (2022). Ideal Prima Pada Daerah Dedekind Berupa Polinom Faktor Berderajat Satu Dari Ring Bilangan Bulat. FRAKTAL: JURNAL MATEMATIKA DAN PENDIDIKAN MATEMATIKA, 3(2), 65-69. https://doi.org/10.35508/fractal.v3i2.8742
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