A Survey on Ring Theory for its Mathematical Foundation and Applications

Authors

  • Huiran Zheng

DOI:

https://doi.org/10.61173/csvw2427

Keywords:

ring theory, algebra, application

Abstract

The origin of the ring theory can be dated to the early 19th century. With the rapid development of technology, more and more complex engineering problems and computer problems need to obtain support and results from more basic mathematics theory. Hence, based on this demand, the researcher discovered that ring theory is a very noteworthy math theory for solving the complicated problems above. This paper investigated the ring theory and related articles based on number theory and corresponding applications, which produces and results in a literature review in this particular direction.

References

Abduldaim, A. M., & Al-Saidi, N. M. (2017). Generalized π-armendariz authentication cryptosystem. International Journal of Mathematical and Computational Sciences, 11 (9), 422–427.

Adhami, R. R., & Brewer, V. E. (1989). Edge detection using a cyclic ring of integers modulo 7. In 1989 the twenty-first southeastern symposium on system theory (pp. 145–146).

Aditya, R., Zulfikar, M. T., & Manik, N. I. (2015). Testing division rings and fields using a computer program. Procedia Computer Science, 59 , 540–549. Ahmed, S., Ghosh, K. K., Singh, P. K., Geem, Z. W., & Sarkar, R. (2020). Hybrid of harmony search algorithm and ring theory-based evolutionary algorithm for feature selection. IEEE Access, 8 , 102629– 102645.

Artin, E. (1927). Zur theorie der hyperkomplexen zahlen. In Abhandlungen aus dem mathematischen seminar der universität hamburg (Vol. 5, pp. 251–260). Awange, J. L., Fukuda, Y., Takemoto, S., & Grafarend, E. W. (2005). Role of algebra in modern day geodesy. In A window on the future of geodesy: Proceedings of the international association of geodesy iag general assembly sapporo, japan june 30–july 11, 2003 (pp. 524–529).

Bell, J., Launois, S., Sánchez, O. L., & Moosa, R. (2017). Poisson algebras via model theory and differential-algebraic geometry. Journal of the European Mathematical Society, 19 (7), 2019–2049.

Blechschmidt, I. (2021). Using the internal language of toposes in algebraic geometry. arXiv preprint arXiv:2111.03685 .

Bosch, S., et al. (2013). Algebraic geometry and commutative algebra. Springer.

Cardarilli, G. C., Nannarelli, A., & Re, M. (2007). Residue number system for low-power dsp applications. In 2007 conference record of the forty-first asilomar conference on signals, systems and computers (pp. 1412–1416).

Chattopadhyay, S., Patra, K. L., & Sahoo, B. K. (2020). Laplacian eigenvalues of the zero divisor graph of the ring zn. Linear Algebra and its applications, 584 , 267–286.

Chervyakov, N., Lyakhov, P., & Valueva, M. (2017). Increasing of convolutional neural network performance using residue number system. In 2017 international multi-conference on engineering, computer and information sciences (sibircon) (pp. 135–140).

Cimprič, J. (2012). Real algebraic geometry for matrices over commutative rings. Journal of Algebra, 359 , 89–103.

Cohn, P. M. (2012). Introduction to ring theory. Springer

Science & Business Media. Connes, A., & Consani, C. (2021). On absolute algebraic geometry the affine case. Advances in Mathematics, 390 , 107909.

Ebrahimi, S., & Bayat-Sarmadi, S. (2020). Lightweight and dpa-resistant post-quantum cryptoprocessor based on binary ring-lwe. In 2020 20th international symposium on computer architecture and digital systems (cads) (pp. 1–6). Ebrahimi, S., Bayat-Sarmadi, S., & Mosanaei-Boorani, H. (2019). Post-quantum cryptoprocessors optimized for edge and resource-constrained devices in iot. IEEE Internet of Things Journal, 6 (3), 5500–5507.

Fraenkel, E. (1914). Über die beziehungen der leukämie zu geschwulstbildenden prozessen des hämatopoetischen apparates. Virchows Archiv für pathologische Anatomie und Physiologie und für klinische Medizin, 216 (3), 340–354.

Galdino, A. L., Borges, A. A., Ayala-Rincón, M., et al. (2021). Formalization of ring theory in pvs. Journal of Automated Reasoning, 65 (8), 1231–1263. Dean&Francis

Garcés, Y., Torres, E., Pereira, O., & Rodríguez, R. (2014). Application of the ring theory in the segmentation of digital images. arXiv preprint arXiv:1402.4069 .

Grigoriev, D., & Ponomarenko, I. (2003). Homomorphic public-key cryptosystems over groups and rings. arXiv preprint cs/0309010 .

Hamilton, W. R. (1843). On quaternions; or on a new system of imaginaries in algebra (letter to john t. graves, dated october 17, 1843). Philos. Magazine, 25 , 489–495.

Harada, M. (1979). Non-small modules and non-cosmall modules. In Ring theory, proceedings of 1978 antwerp conference.

He, Y., Wang, X., & Gao, S. (2019). Ring theory-based evolutionary algorithm and its application to d {0-1} kp. Applied Soft Computing, 77 , 714– 722.

Jacobson, N. (1956). Structure of rings (Vol. 37). American Mathematical Soc.

Jiang, Y., & Zhang, Y. (2018). Algebraic geometry and bethe ansatz. part i. the quotient ring for bae. Journal of High Energy Physics, 2018 (3), 1–40.

Khan, A. (2019). The morel–voevodsky localization theorem in spectral algebraic geometry. Geometry & Topology, 23 (7), 3647–3685.

Kleiner, I. (1996). The genesis of the abstract ring concept. The American mathematical monthly, 103 (5), 417–424.

Kleiner, I. (1998). From numbers to rings: The early history of ring theory. Elemente der Mathematik, 53 , 18–35.

Knapp, A. W. (2007). Advanced algebra. Springer Science & Business Media.

Kolmogorov, A. N. (1900). Kolmogorov 1903–1987. In Proc. 3rd ieee conference on structure in (pp. 80–101).

Krull, W. (2020). Primidealketten in allgemeinen ringbereichen. Walter de Gruyter GmbH & Co KG.

Lemmermeyer, F. (2011). The snake lemma. arXiv preprint arXiv:1108.5684 .

Lezama, O., & Latorre, E. (2017). Non-commutative algebraic geometry of semi-graded rings. International Journal of Algebra and Computation, 27 (04), 361–389.

Lyubashevsky, V., Peikert, C., & Regev, O. (2010). On ideal lattices and learning with errors over rings. In Advances in cryptology–eurocrypt 2010: 29th annual international conference on the theory and applications of cryptographic techniques, french riviera, may 30–june 3, 2010. proceedings 29 (pp. 1–23). Malcev, A. I. (n.d.). 1909–1967. Muthuraj, R., & Gandhi, N. R. (n.d.). Application of hx ring theory in homomorphic encryption. Noether, E. (n.d.). Ideal theory in rings (idealtheorie in ringbereichen), translated by daniel berlyne. arXiv preprint arXiv:1401.2577 .

Noether, E. (1923). Eliminationstheorie und allgemeine idealtheorie. Mathematische Annalen, 90 (3-4), 229–261.

Pirzada, S., Rather, B., Shaban, R. U., & Merajuddin, S. (2021). On signless laplacian spectrum of the zero divisor graphs of the ring zn. Korean Journal of Mathematics, 29 (1), 13–24.

Pirzada, S., Rather, B. A., & Chishti, T. (2021). On distance laplacian spectrum of zero divisor graphs of the ring zn. Carpathian Mathematical Publications, 13 (1), 48–57.

Rowen, L. H. (2012). Ring theory, 83. Academic Press.

Sarathy, R., & Sankar, J. R. (2023). Applications on color (distance) signless laplacian energy of annihilator monic prime graph of commutative rings. Ain Shams Engineering Journal, 102469.

Shariq, M., Mathil, P., & Kumar, J. (2023). Laplacian spectrum of weakly zero-divisor graph of the ring zn. arXiv preprint arXiv:2307.12757 .

Shen, S., Liu, W., & Jin, W. (2023). Laplacian eigenvalues of the unit graph of the ring zn. Applied Mathematics and Computation, 459 , 128268.

Shirshov, A. I. (1962). On the bases of a free lie algebra. Algebra

i Logika, 1 (1), 14–19. Stillwell, J., & Stillwell, J. (1989). Mathematics and its history (Vol. 3). Springer. Torres, E., Rodriguez, R., Garcés, Y., & Pereira, O. (n.d.). Edge detection in segmented images through mean shift iterative gradient using ring. Valueva, M. V., Nagornov, N., Lyakhov, P. A., Valuev, G. V., &

Chervyakov, N. I. (2020). Application of the residue number system to reduce hardware costs of the convolutional neural network implementation. Mathematics and computers in simulation, 177 , 232–243.

Wedderburn, J. M. (1908). On hypercomplex numbers. Proceedings of the London Mathematical Society, 2 (1), 77–118.

Zheng, Z., Huang, W., Xu, J., & Tian, K. (2021). A generalization of cyclic code and applications to public key cryptosystems. arXiv preprint arXiv:21

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Published

2024-10-29