| Title:
|
Gaussian and Prüfer conditions in bi-amalgamated algebras (English) |
| Author:
|
Mahdou, Najib |
| Author:
|
Moutui, Moutu Abdou Salam |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
2 |
| Year:
|
2020 |
| Pages:
|
381-391 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $f\colon A\rightarrow B$ and $g\colon A\rightarrow C$ be two ring homomorphisms and let $J$ and $J'$ be ideals of $B$ and $C$, respectively, such that $f^{-1}(J)=g^{-1}(J')$. In this paper, we investigate the transfer of the notions of Gaussian and Prüfer rings to the bi-amalgamation of $A$ with $(B,C)$ along $(J,J')$ with respect to $(f,g)$ (denoted by $A\bowtie ^{f,g}(J,J')),$ introduced and studied by S. Kabbaj, K. Louartiti and M. Tamekkante in 2013. Our results recover well known results on amalgamations in C. A. Finocchiaro (2014) and generate new original examples of rings possessing these properties. (English) |
| Keyword:
|
bi-amalgamation |
| Keyword:
|
amalgamated algebra |
| Keyword:
|
Gaussian ring |
| Keyword:
|
Prüfer ring |
| MSC:
|
13A15 |
| MSC:
|
13C10 |
| MSC:
|
13C11 |
| MSC:
|
13D05 |
| MSC:
|
13E05 |
| MSC:
|
13F05 |
| MSC:
|
13F20 |
| MSC:
|
13F30 |
| MSC:
|
16D40 |
| MSC:
|
16E10 |
| MSC:
|
16E60 |
| idZBL:
|
07217141 |
| idMR:
|
MR4111849 |
| DOI:
|
10.21136/CMJ.2019.0335-18 |
| . |
| Date available:
|
2020-06-17T12:32:19Z |
| Last updated:
|
2022-07-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148235 |
| . |
| Reference:
|
[1] Abuhlail, J., Jarrar, M., Kabbaj, S.: Commutative rings in which every finitely generated ideal is quasi-projective.J. Pure Appl. Algebra 215 (2011), 2504-2511. Zbl 1226.13014, MR 2793953, 10.1016/j.jpaa.2011.02.008 |
| Reference:
|
[2] Bakkari, C., Kabbaj, S., Mahdou, N.: Trivial extensions defined by Prüfer conditions.J. Pure Appl. Algebra 214 (2010), 53-60. Zbl 1175.13008, MR 2561766, /10.1016/j.jpaa.2009.04.011 |
| Reference:
|
[3] Bazzoni, S., Glaz, S.: Prüfer rings.Multiplicative Ideal Theory in Commutative Algebra J. W. Brewer et al. Springer, New York (2006), 55-72. Zbl 1115.13024, MR 2265801, 10.1007/978-0-387-36717-0_4 |
| Reference:
|
[4] Bazzoni, S., Glaz, S.: Gaussian properties of total rings of quotients.J. Algebra 310 (2007), 180-193. Zbl 1118.13020, MR 2307788, 10.1016/j.jalgebra.2007.01.004 |
| Reference:
|
[5] Boisen, M. B., Sheldon, P. B.: CPI-extension: Overrings of integral domains with special prime spectrums.Can. J. Math. 29 (1977), 722-737. Zbl 0363.13002, MR 0447205, 10.4153/CJM-1977-076-6 |
| Reference:
|
[6] Butts, H. S., Smith, W.: Prüfer rings.Math. Z. 95 (1967), 196-211. Zbl 0153.37003, MR 0209271, 10.1007/BF01111523 |
| Reference:
|
[7] Campanini, F., Finocchiaro, C. A.: Bi-amalgamated constructions.J. Algebra Appl. 18 (2019), Article ID 1950148, 16 pages. Zbl 07096463, MR 3977809, 10.1142/S0219498819501482 |
| Reference:
|
[8] Chhiti, M., Jarrar, M., Kabbaj, S., Mahdou, N.: Prüfer conditions in an amalgamated duplication of a ring along an ideal.Commun. Algebra 43 (2015), 249-261. Zbl 1327.13063, MR 3240418, 10.1080/00927872.2014.897575 |
| Reference:
|
[9] D'Anna, M.: A construction of Gorenstein rings.J. Algebra 306 (2006), 507-519. Zbl 1120.13022, MR 2271349, 10.1016/j.jalgebra.2005.12.023 |
| Reference:
|
[10] D'Anna, M., Finocchiaro, C. A., Fontana, M.: Amalgamated algebras along an ideal.Commutative Algebra and its Applications M. Fontana et al. Walter de Gruyter, Berlin (2009), 155-172. Zbl 1177.13043, MR 2606283, 10.1515/9783110213188.155 |
| Reference:
|
[11] D'Anna, M., Finocchiaro, C. A., Fontana, M.: Properties of chains of prime ideals in an amalgamated algebra along an ideal.J. Pure Appl. Algebra 214 (2010), 1633-1641. Zbl 1191.13006, MR 2593689, 10.1016/j.jpaa.2009.12.008 |
| Reference:
|
[12] D'Anna, M., Fontana, M.: An amalgamated duplication of a ring along an ideal: The basic properties.J. Algebra Appl. 6 (2007), 443-459. Zbl 1126.13002, MR 2337762, 10.1142/S0219498807002326 |
| Reference:
|
[13] Finocchiaro, C. A.: Prüfer-like conditions on an amalgamated algebra along an ideal.Houston J. Math. 40 (2014), 63-79. Zbl 1297.13002, MR 3210554 |
| Reference:
|
[14] Fuchs, L.: Über die Ideale arithmetischer Ringe.Comment. Math. Helv. 23 (1949), 334-341 German. Zbl 0040.30103, MR 0032583, 10.1007/BF02565607 |
| Reference:
|
[15] Glaz, S.: Prüfer conditions in rings with zero-divisors.Arithmetical Properties of Commutative Rings and Monoids S. T. Chapman Lecture Notes in Pure Applied Mathematics 241, Chapman & Hall/CRC, Boca Raton (2005), 272-281. Zbl 1107.13023, MR 2140700, 10.1201/9781420028249.ch17 |
| Reference:
|
[16] Glaz, S.: The weak global dimensions of Gaussian rings.Proc. Am. Math. Soc. 133 (2005), 2507-2513. Zbl 1077.13009, MR 2146192, 10.1090/S0002-9939-05-08093-7 |
| Reference:
|
[17] Griffin, M.: Prüfer rings with zero divisors.J. Reine Angew. Math. 239-240 (1969), 55-67. Zbl 0185.09801, MR 0255527, 10.1515/crll.1969.239-240.55 |
| Reference:
|
[18] Kabbaj, S., Louartiti, K., Tamekkante, M.: Bi-amalgamated algebras along ideals.J. Commut. Algebra 9 (2017), 65-87. Zbl 1390.13008, MR 3631827, 10.1216/JCA-2017-9-1-65 |
| Reference:
|
[19] Kabbaj, S., Mahdou, N., Moutui, M. A. S.: Bi-amalgamations subject to the arithmetical property.J. Algebra Appl. 16 (2017), Article ID 1750030, 11 pages. Zbl 1390.13057, MR 3608417, 10.1142/S021949881750030X |
| Reference:
|
[20] Koehler, A.: Rings for which every cyclic module is quasi-projective.Math. Ann. 189 (1970), 311-316. Zbl 0198.05602, MR 0280536, 10.1007/BF01359713 |
| Reference:
|
[21] Krull, W.: Beiträge zur Arithmetik kommutativer Integritätsbereiche.Math. Z. 41 (1936), 545-577 German. Zbl 0015.00203, MR 1545640, 10.1007/BF01180441 |
| Reference:
|
[22] Loper, K. A., Roitman, M.: The content of a Gaussian polynomial is invertible.Proc. Am. Math. Soc. 133 (2005), 1267-1271. Zbl 1137.13301, MR 2111931, 10.1090/S0002-9939-04-07826-8 |
| Reference:
|
[23] Lucas, T. G.: Gaussian polynomials and invertibility.Proc. Am. Math. Soc. 133 (2005), 1881-1886. Zbl 1154.13301, MR 2137851, 10.1090/S0002-9939-05-07977-3 |
| Reference:
|
[24] Prüfer, H.: Untersuchungen über Teilbarkeitseigenschaften in Körpern.J. Reine Angew. Math. 168 (1932), 1-36 German. Zbl 0004.34001, MR 1581355, 10.1515/crll.1932.168.1 |
| Reference:
|
[25] Tsang, H.: Gauss's Lemma, Ph.D. Thesis.University of Chicago, Chicago (1965). MR 2611536 |
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