Öz
Kaynakça
- J. A. Gallian, Contemporary Abstract Algebra, 5th edition, Houghton Mifflin Co., Boston, 1998.
- J. T. Cross, The Euler’s φ-function in the Gaussian Integers, American Mathematical Monthly Journal 55 (1983) 518–528.
- J. L. Smith, J. A. Gallian, Factoring Finite Factor Rings, Mathematics Magazine 58 (1985) 93–95.
- A. N. El-Kassar, H. Y. Chehade, D. Zatout, Quotient Rings of Polynomials over Finite Fields with Cyclic Groups of units, Proceedings of the International Conference on Research Trends in Science and Technology, RTST2002, Lebanese American University, Beirut Lebanon (2002) 257–266.
- A. N. El-Kassar, H. Y. Chehade, Generalized Group of Units, Mathematica Balkanica, New Series 20 (2006) 275–286.
- A. A. Allan, M. J. Dunne, J. R. Jack, J. C. Lynd, H. W. Ellingsen, Classification of the Group of Units in the Gaussian Integers Modulo N*, Pi Mu Epsilon Journal 12 (9) (2008) 513–519.
- W. K. Buck, Cyclic Rings, Master’s Thesis, Eastern Illinois University: The Keep (2004).
- T. W. Hungerford, Algebra: Graduate Texts in Mathematics, Springer, New York, 2003.
- W. M. Fakieh, S. K. Nauman, Reversible Rings with Involutions and Some Minimalities, The Scientific World Journal, Hindawi Publishing Corporation 2013 (2013) 1–8.
- I. N. Herstein, S. Montgomery, A Note on Division Rings with Involutions, Michigan Mathematical Journal 18 (1) (1971) 75–79.
- D. I. C. Mendes, A Note on Involution Rings, Miskolc Mathematical Notes 10 (2) (2009) 155–162.
- W. M. Fakieh, Symmetric Rings with Involutions, British Journal of Mathematics and Computer Science 8 (6) (2015) 492–505.
- T. Chalapathi, R. V. M. S. S. K. Kumar, Self-Additive Inverse Elements of Neutrosophic Rings and Fields, Annals of Pure and Applied Mathematics 13 (1) (2017) 63–72.
- V. K. Khanna, S. K. Bhambri, A Course in Abstract Algebra, 2nd Edition, Vikas Publishing House Pvt. Ltd, 1998.
- D. Suzanne, How Many Solutions Does x^2+1=0 Have? An Abstract Algebra Project, PRIMUS Journal 10 (2) (2007) 111–122.
- G. Dresden, W. M. Dymacek, Finding Factors of Factor Rings over the Gaussian Integers, The American Mathematical Monthly Journal 112 (7) (2018) 602–611.
- T. Andreescu, D. Andrica, I. Cucurezean, An Introduction to Diophantine Equations: A Problem-Based Approach, Springer, New York, 2010.
- T. M. Apostol, Introduction to Analytic Number Theory, Springer International Student Edition, 1989.
Öz
In this paper, we study a special class of elements in the finite commutative rings called involutions. An involution of a ring R is an element with the property that x^2-1=0 for some x in R. This study describes both the implementation and enumeration of the involutions of various rings, such as cyclic rings, non-cyclic rings, zero-rings, finite fields, and especially rings of Gaussian integers. The paper begins with simple well-known results of an equation x^2-1=0 over the finite commutative ring R. It provides a concrete setting to enumerate the involutions of the finite cyclic and non-cyclic rings R, along with the isomorphic relation I(R)≅Z_2^k.
Anahtar Kelimeler
Cyclic rings noncyclic rings zero rings finite fields involutions
Kaynakça
- J. A. Gallian, Contemporary Abstract Algebra, 5th edition, Houghton Mifflin Co., Boston, 1998.
- J. T. Cross, The Euler’s φ-function in the Gaussian Integers, American Mathematical Monthly Journal 55 (1983) 518–528.
- J. L. Smith, J. A. Gallian, Factoring Finite Factor Rings, Mathematics Magazine 58 (1985) 93–95.
- A. N. El-Kassar, H. Y. Chehade, D. Zatout, Quotient Rings of Polynomials over Finite Fields with Cyclic Groups of units, Proceedings of the International Conference on Research Trends in Science and Technology, RTST2002, Lebanese American University, Beirut Lebanon (2002) 257–266.
- A. N. El-Kassar, H. Y. Chehade, Generalized Group of Units, Mathematica Balkanica, New Series 20 (2006) 275–286.
- A. A. Allan, M. J. Dunne, J. R. Jack, J. C. Lynd, H. W. Ellingsen, Classification of the Group of Units in the Gaussian Integers Modulo N*, Pi Mu Epsilon Journal 12 (9) (2008) 513–519.
- W. K. Buck, Cyclic Rings, Master’s Thesis, Eastern Illinois University: The Keep (2004).
- T. W. Hungerford, Algebra: Graduate Texts in Mathematics, Springer, New York, 2003.
- W. M. Fakieh, S. K. Nauman, Reversible Rings with Involutions and Some Minimalities, The Scientific World Journal, Hindawi Publishing Corporation 2013 (2013) 1–8.
- I. N. Herstein, S. Montgomery, A Note on Division Rings with Involutions, Michigan Mathematical Journal 18 (1) (1971) 75–79.
- D. I. C. Mendes, A Note on Involution Rings, Miskolc Mathematical Notes 10 (2) (2009) 155–162.
- W. M. Fakieh, Symmetric Rings with Involutions, British Journal of Mathematics and Computer Science 8 (6) (2015) 492–505.
- T. Chalapathi, R. V. M. S. S. K. Kumar, Self-Additive Inverse Elements of Neutrosophic Rings and Fields, Annals of Pure and Applied Mathematics 13 (1) (2017) 63–72.
- V. K. Khanna, S. K. Bhambri, A Course in Abstract Algebra, 2nd Edition, Vikas Publishing House Pvt. Ltd, 1998.
- D. Suzanne, How Many Solutions Does x^2+1=0 Have? An Abstract Algebra Project, PRIMUS Journal 10 (2) (2007) 111–122.
- G. Dresden, W. M. Dymacek, Finding Factors of Factor Rings over the Gaussian Integers, The American Mathematical Monthly Journal 112 (7) (2018) 602–611.
- T. Andreescu, D. Andrica, I. Cucurezean, An Introduction to Diophantine Equations: A Problem-Based Approach, Springer, New York, 2010.
- T. M. Apostol, Introduction to Analytic Number Theory, Springer International Student Edition, 1989.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik |
| Bölüm | Araştırma Makalesi |
| Yazarlar | |
| Yayımlanma Tarihi | 30 Eylül 2021 |
| Gönderilme Tarihi | 15 Temmuz 2021 |
| Yayımlandığı Sayı | Yıl 2021 Sayı: 36 |
Kaynak Göster
- Makale Dosyaları