Abstract
Marks (J. Algebra 280 (2004) 463–471) proved that if the skew polynomial ring \(R[x;\sigma ]\) is left or right duo, then \(R[x;\sigma ]\) is commutative. It is proved that if \(R[x;\sigma ]\) is weakly left (resp., right) duo over a reduced ring R with an endomorphism (resp., a monomorphism) \(\sigma \), then \(R[x;\sigma ]\) is commutative. This concludes that a noncommutative skew polynomial ring is not weakly left duo when the base ring is reduced. It is also shown that if \(R[x;\sigma ]\) is weakly left duo then the polynomial ring R[x] is weakly left duo. We next study the structure of the Ore extension \(R[x; \sigma ,\delta ]\) when it is weakly left or right duo.
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Acknowledgements
The authors would like to thank the referee for very careful reading of the manuscript and valuable suggestions in depth that improved the paper much. The third-named author was supported by the research fund of Hanbat National University in 2018.
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Communicating Editor: Mrinal Kanti Das
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Hong, C.Y., Kim, H.K., Kim, N.K. et al. Structure of weakly one-sided duo Ore extensions. Proc Math Sci 131, 3 (2021). https://doi.org/10.1007/s12044-020-00600-9
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DOI: https://doi.org/10.1007/s12044-020-00600-9
Keywords
- Weakly left (right) duo ring
- skew polynomial ring
- ore extension
- rigid endomorphism
- commutative ring
- radical