Abstract
Tarizadeh and Aghajani conjectured that each purely-prime ideal is purely-maximal (Tarizadeh and Aghajani in Commun Algebra 49(2):824–835, 2021, Conjecture 5.8). We study purely-prime and purely-maximal ideals in rings of the form \(A+XS\) (where S is either B[X] or B[[X]]), subrings of A[[X]] of the form \(A[X]+I[[X]]\) and \(A+I[[X]]\) (where A is a subring of a commutative unitary ring B and I an ideal of A) and Nagata’s idealization ring. As application, we give necessary and sufficient conditions on each of the aforementioned ring to be semi-Noetherian. We deduce that the power series ring A[[X]] is semi-Noetherian if and only if the ring A is semi-Noetherian. We deduce that Tarizadeh and Aghajani’s conjecture holds in each of the aforementioned ring if and only if it holds in the ring A.
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Acknowledgements
The authors would like to thank the referee for several valuable comments which suggested some alternate proofs and additional examples. We have incorporated several of referee’s suggestions in the paper and especially Theorem 2.14 is due to the referee. The first author is grateful to Dr. Mohamed Khalifa for valuable discussions.
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Ouni, N., Benhissi, A. On purely-maximal ideals and semi-Noetherian power series rings. Beitr Algebra Geom 65, 229–240 (2024). https://doi.org/10.1007/s13366-023-00685-z
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DOI: https://doi.org/10.1007/s13366-023-00685-z