| Title:
|
Notes on generalizations of Bézout rings (English) |
| Author:
|
El Alaoui, Haitham |
| Author:
|
Mouanis, Hakima |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
62 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
265-272 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we give new characterizations of the $P$-$2$-Bézout property of trivial ring extensions. Also, we investigate the transfer of this property to homomorphic images and to finite direct products. Our results generate original examples which enrich the current literature with new examples of non-$2$-Bézout $P$-$2$-Bézout rings and examples of non-$P$-Bézout $P$-$2$-Bézout rings. (English) |
| Keyword:
|
$P$-Bézout ring |
| Keyword:
|
2-Bézout ring |
| Keyword:
|
$P$-2-Bézout ring |
| Keyword:
|
trivial rings extension |
| Keyword:
|
homomorphic image |
| Keyword:
|
finite direct product |
| MSC:
|
13A15 |
| MSC:
|
13B10 |
| MSC:
|
13D02 |
| MSC:
|
13F05 |
| idMR:
|
MR4331281 |
| DOI:
|
10.14712/1213-7243.2021.020 |
| . |
| Date available:
|
2021-10-18T08:27:10Z |
| Last updated:
|
2023-10-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149142 |
| . |
| Reference:
|
[1] Anderson D. D., Winders M.: Idealization of a module.J. Commut. Algebra 1 (2009), no. 1, 3–56. Zbl 1194.13002, 10.1216/JCA-2009-1-1-3 |
| Reference:
|
[2] Bakkari C.: On $P$-Bézout rings.Int. J. of Algebra 3 (2009), no. 13–16, 669–673. |
| Reference:
|
[3] Bakkari C.: $P$-Bézout property in pullbacks.Int. J. Contemp. Math. Sci. 4 (2009), no. 21–24, 1209–1213. |
| Reference:
|
[4] Bakkari C., Ouarghi K.: On $2$-Bézout rings.Int. J. Algebra 4 (2010), no. 5–8, 241–245. |
| Reference:
|
[5] Costa D. L.: Parameterizing families of non-Noetherian rings.Comm. Algebra 22 (1994), no. 10, 3997–4011. Zbl 0814.13010, 10.1080/00927879408825061 |
| Reference:
|
[6] El Alaoui H., Mouanis H.: On $P$-$2$-Bézout rings.Gulf J. Math. 7 (2019), no. 3, 1–6. |
| Reference:
|
[7] Gillman L., Henriksen M.: Some remarks about elementary divisor rings.Trans. Amer. Math. Soc. 82 (1956), 362–365. 10.1090/S0002-9947-1956-0078979-8 |
| Reference:
|
[8] Glaz S.: Commutative Coherent Rings.Lecture Notes in Mathematics, 1371, Springer, Berlin, 1989. Zbl 0787.13001, 10.1007/BFb0084576 |
| Reference:
|
[9] Huckaba J. A.: Commutative Rings with Zero Divisors.Monographs and Textbooks in Pure and Applied Mathematics, 117, Marcel Dekker, New York, 1988. Zbl 0637.13001 |
| Reference:
|
[10] Kabbaj S.-E., Mahdou N.: Trivial extensions defined by coherent-like conditions.Comm. Algebra 32 (2004), no. 10, 3937–3953. Zbl 1068.13002, 10.1081/AGB-200027791 |
| Reference:
|
[11] Kaplansky I.: Elementary divisors and modules.Trans. Amer. Math. Soc. 66 (1949), 464–491. 10.1090/S0002-9947-1949-0031470-3 |
| Reference:
|
[12] Larsen M. D., Lewis W. J., Shores T. S.: Elementary divisors rings and finitely presented modules.Trans. Amer. Math. Soc. 187 (1974), 231–248. 10.1090/S0002-9947-1974-0335499-1 |
| Reference:
|
[13] Mahdou N.: On Costa's conjecture.Comm. Algebra 29 (2001), no. 7, 2775–2785. 10.1081/AGB-4986 |
| Reference:
|
[14] Mahdou N.: On $2$-von Neumann regular rings.Comm. Algebra 33 (2005), no. 10, 3489–3496. Zbl 1080.13004, 10.1080/00927870500242991 |
| Reference:
|
[15] Palmér I., Roos J.-E.: Explicit formulae for the global homological dimensions of trivial extensions of rings.J. Algebra 27 (1973), 380–413. 10.1016/0021-8693(73)90113-0 |
| Reference:
|
[16] Rotman J. J.: An Introduction to Homological Algebra.Pure and Applied Mathematics, 85, Academic Press, Harcourt Brace Jovanovich, New York, 1979. Zbl 1157.18001 |
| . |
