A well-known result by Zariski and Samuel asserts that a commutative principal ideal ring is a direct sum of finitely many principal ideal domains and Artinian chain rings. Based on this result, it is shown, among other things, that a commutative polynomial ring R[x] is a principal ideal ring if and only if R is a finite direct sum of fields.
- Principal ideal ring
- Polynomial ring
- Principal ideal domain
- Artinian chain ring
- Bézout domain
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The author gratefully thanks to the anonymous referees for their valuable comments which substantially improved the presentation of the paper.
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Chimal-Dzul, H. (2022). Commutative Polynomial Rings which are Principal Ideal Rings. In: Ashraf, M., Ali, A., De Filippis, V. (eds) Algebra and Related Topics with Applications. ICARTA 2019. Springer Proceedings in Mathematics & Statistics, vol 392. Springer, Singapore. https://doi.org/10.1007/978-981-19-3898-6_7
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