- Presimplifiable Ring,
- Weakly Presimplifiable Ring,
- Polynomial Ring,
- Formal Power Series Ring
Copyright (c) 2023 Deby Anastasya, Sri Wahyuni
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Abstract
Let be a commutative ring with identity. Two elements and b in are called to be associates if and , or equivalently, if . The generalization of associate relation in R has given the idea for definitions of presimplifiable and weakly presimplifiable rings. First of all, it will be given definitions of very strong associate relation, strong regular associate relation, very strongly associate ring, and strongly regular associate ring. The presimplifiable ring is a commutative ring with the condition that every nonzero element is a unit element. While the weakly presimplifiable ring is a commutative ring with the condition that every nonzero element is regular element. Furthermore, it is shown that the relationship between very strongly associate ring with presimplifiable ring and the linkage between strongly regular associate ring and weakly presimplifiable ring. It is obtained that is a presimplifiable ring if and only if is a very strongly associate ring. Meanwhile, is a weakly presimplifiable ring if and only if is a strongly regular associate ring. Then, it is shown that the correlation between presimplifiable and weakly presimplifiable rings to its polynomial ring and its the formal power series ring . If is a weakly presimplifiable ring, then and are also weakly presimplifiable rings. However, if is a presimplifiable ring, then is also a presimplifiable ring but always not valid for .
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References
- A. Alsaraireh, “Presimplifiable Rings and Modules,” Research India Publications, vol. 8, no. 1, pp. 33-37, 2015.
- C.P. Mooney and Y. Wang, “On τ-U-irreducible elements in Strongly Associate Ring,” The Minnesota Journal of
- Undergraduate Mathematics, vol. 4, pp. 1-15, 2018.
- D. D. Anderson and S. Valdes-Leon, “Factorization in Commutative Rings with Zero Divisors,” Rocky Mountain J. Math.,
- Vol. 2, no. 26, pp. 439-480, 1996.
- D. D. Anderson and S. Valdes-Leon, “Factorization in Commutative Rings with Zero Divisors, II, Factorization in Integral
- Domain,” Lecture Notes in Pure and Appl. Math., pp. 197-219, 1997.
- D. D. Anderson, M. Axtell, S. J. Forman, and J. Stickles, “When are Associates Unit Multiples?,” Rocky Mountain J. Math.,
- vol. 34, no. 3, pp. 811-828, 2004.
- D. D. Anderson and M. R. Winders, “Idealization of a Module,” Journal of Commutative Algebra, vol. 1, no. 1, pp. 3-56,
- D. D. Anderson and S. Chun, “Associate Elements in Commutative Rings,” Rocky Mountain J. Math., vol. 44, no. 3, pp.
- -731, 2014.
- D. S. Malik, J. M. Mordeson, and M. K. Sen, Fundamentals of Abstract Algebra, Singapore: McGraw-Hill Companies,
- D. Spellman, G. M. Benkart, A. M. Gaglione, M. E. Kidwell, M. D. Meyerson, and W. P. Wardlaw, “Principal Ideals and
- Associate Rings,” JP. Journal of Algebra, Number Theory and Applications, vol. 2, pp. 181-193, 2002.
- H. A. Khashan and E. Y. Celikel, “Weakly J-ideals of Commutative Rings,” Filomat, vol. 36, no. 2, pp. 485-495, 2022.
- I. Kaplansky, “Elementary Divisors and Modules,” Tranc. Amer. Math. Soc., vol. 66, pp. 464-491, 1949.
- M. Ghanem, “Some Properties of Associate and Presimplifiable Rings,” Turk J. Math., vol. 35, pp. 333-340, 1949.
- P. M. Cohn, Introduction to Ring Theory, London: Springer, 2000.
- S. Roman, Advanced Linear Algebra Third Edition, New York: Springer, 2008.
- W. A. Adkins and S. H. Weintraub, An Approach via Module Theory, New York: Springer-Verlag, 1992.