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Rings in which every regular ideal is projective | ||
| Journal of Algebra and Related Topics | ||
| دوره 14، Special Issue- Dedicated to the memory of Jürgen Herzog (1941-2024).، تیر 2026، صفحه 183-190 اصل مقاله (146.61 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2025.27374.1662 | ||
| نویسنده | ||
| S. E. Mahdou* | ||
| Laboratory of Modelling and Mathematical Structures, Faculty of Science and Technology of Fez, University S. M. Ben Abdellah Fez, Morocco | ||
| چکیده | ||
| In this paper, we introduce a new class of ring called regular hereditary ring, which is a weak version of hereditary ring property. Any hereditary ring is naturally a regular hereditary ring, and in the domain context, these two forms coincide to become a Dedekind domain. We study the transfer of this notion to various context of commutative ring extensions such as localization, direct product, trivial ring extensions and pullbacks. Our results generate new families of examples of non-hereditary regular hereditary rings. | ||
| کلیدواژهها | ||
| Regular ideal؛ Regular hereditary ring؛ Trivial ring extension؛ Pullbacks | ||
| مراجع | ||
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[1] D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra, (1) 1,(2009), 3–56. [2] E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form D + M, Michigan Math. J., 20 (1973), 79–95. [3] J. W. Brewer and E. A. Rutter, D+M constructions with general overrings, Michigan Math. J., 23 (1976), 33–42. [4] M. Chhiti and S. E. Mahdou, S-Coherent property in trivial extension and in amalgamation, Commun. Korean Math. Soc., (3) 38 (2023), 705-714. [5] D. E. Dobbs, A. El Khalfi and N. Mahdou, Trivial extensions satisfying certain valuation-like properties, Commun. Algebra, (5) 47, (2019), 2060–2077. [6] T. Dumitrescu, N. Mahdou and Y. Zahir, Radical factorization for trivial extensions and amalgamated duplication rings, J. Algebra Appl., (2) 20, (2021), 2150025, 10 pp. [7] R. El Khalfaoui and N. Mahdou, The ϕ-Krull dimension of some commutative extensions, Commun. Algebra, (9) 48, (2020), 3800-3810. [8] S. Glaz, Commutative Coherent Rings, Lecture Notes in Math, 1371, (1989), Springer-Verlag, Berlin. [9] J. A. Huckaba, Commutative Rings with Zero Divisors, (1988), Dekker, New York. [10] S. Kabbaj, Matlis′ semi-regularity and semi-coherence in trivial ring extensions: a survey, Moroccan Journal of Algebra and Geometry with Applications, (1) 1 (2022), 1-17. [11] I. Kaplansky, Elementary divisors and modules, Proc. Amer. Math. Soc., 66 (1949), 464-491. [12] M. Issoual and N. Mahdou, Trivial Extensions defined by 2-absorbing-like conditions, J. Algebra Appl., (11) 17 (2018), 1850208, 10 pp. [13] S. Kabbaj and N. Mahdou, Trivial extensions defined by coherent-like conditions, Commun. Algebra, (1) 32 (2004), 3937–3953. [14] M. Kabbour, N. Mahdou and A. Mimouni, Trivial ring extensions defined by arithmeticallike properties, Commun. Algebra, (12) 41 (2013), 4534-4548. [15] N. Mahdou, On Cost’as conjecture, Commun. Algebra, (7) 29 (2001), 2775–2785. [16] N. Mahdou, M. A. S. Moutui and Y. Zahir, On S-weakly prime ideals of commutative rings, Georgian Mathematical Journal, (3) 29 (2023), 497–405. [17] M. A. S. Moutui and N. Ouled Azaiez,Some commutative ring extensions defined by almost Bézout condition, Hacet. J. Math. Stat., (1) 49 (2020), 371-379 | ||
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آمار تعداد مشاهده مقاله: 164 تعداد دریافت فایل اصل مقاله: 71 |
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