| Autor: |
Ouni, Nader, Benhissi, Ali |
| Zdroj: |
Contributions to Algebra & Geometry; Mar2024, Vol. 65 Issue 1, p229-240, 12p |
| Abstrakt: |
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 + X S (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. [ABSTRACT FROM AUTHOR] |
| Databáze: |
Complementary Index |
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