Research and Innovation Articles
Published 2018-03-06
Keywords
- Principal ideal ring,
- polynomial ring,
- finite rings
How to Cite
Chimal-Dzul, H., & López-Andrade, C. A. (2018). When is R[x] a principal ideal ring?. Revista Integración, Temas De matemáticas, 35(2), 143–148. https://doi.org/10.18273/revint.v35n2-2017001
Copyright (c) 2018 Revista Integración, temas de matemáticas
This work is licensed under a Creative Commons Attribution 4.0 International License.
Abstract
Because of its interesting applications in coding theory, cryptography, and algebraic combinatoris, in recent decades a lot of attention has been paid to the algebraic structure of the ring of polynomials R[x], where R is a finite commutative ring with identity. Motivated by this popularity, in this paper we determine when R[x] is a principal ideal ring. In fact, we prove that R[x] is a principal ideal ring if and only if R is a finite direct product of finite fields
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References
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[3] Gómez-Calderón J. and Mullen G. L., "Galois rings and algebraic cryptography", Acta Arith. 59 (1991), No. 4, 317-328.
[4] Lidl R. and Niederreiter H., Finite Fields, Cambridge University Press, Cambridge, 1997.
[5] McDonald B., Finite rings with identity, Marcel Dekker, New York, 1974.
[6] Xiang-dong H.A. and Nechaev A.A., "A construction of fnite Frobenius Rings and its application to partial difference sets", J. Algebra 309 (2007), No. 1, 1-9.
[7] Zariski O. and Samuel P., Commutative Algebra I, Springer-Verlag, New York, 1975.