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Gaussian and Prüfer conditions in bi-amalgamated algebras
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 2, pp. 381-391
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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.
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.
DOI : 10.21136/CMJ.2019.0335-18
Classification : 13A15, 13C10, 13C11, 13D05, 13E05, 13F05, 13F20, 13F30, 16D40, 16E10, 16E60
Keywords: bi-amalgamation; amalgamated algebra; Gaussian ring; Prüfer ring 
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Mahdou, Najib; Moutui, Moutu Abdou Salam. Gaussian and Prüfer conditions in bi-amalgamated algebras. Czechoslovak Mathematical Journal, Tome 70 (2020) no. 2, pp. 381-391. doi: 10.21136/CMJ.2019.0335-18

[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. | DOI | MR | JFM

[2] Bakkari, C., Kabbaj, S., Mahdou, N.: Trivial extensions defined by Prüfer conditions. J. Pure Appl. Algebra 214 (2010), 53-60. | DOI | MR | JFM

[3] Bazzoni, S., Glaz, S.: Prüfer rings. Multiplicative Ideal Theory in Commutative Algebra J. W. Brewer et al. Springer, New York (2006), 55-72. | DOI | MR | JFM

[4] Bazzoni, S., Glaz, S.: Gaussian properties of total rings of quotients. J. Algebra 310 (2007), 180-193. | DOI | MR | JFM

[5] Boisen, M. B., Sheldon, P. B.: CPI-extension: Overrings of integral domains with special prime spectrums. Can. J. Math. 29 (1977), 722-737. | DOI | MR | JFM

[6] Butts, H. S., Smith, W.: Prüfer rings. Math. Z. 95 (1967), 196-211. | DOI | MR | JFM

[7] Campanini, F., Finocchiaro, C. A.: Bi-amalgamated constructions. J. Algebra Appl. 18 (2019), Article ID 1950148, 16 pages. | DOI | MR | JFM

[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. | DOI | MR | JFM

[9] D'Anna, M.: A construction of Gorenstein rings. J. Algebra 306 (2006), 507-519. | DOI | MR | JFM

[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. | DOI | MR | JFM

[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. | DOI | MR | JFM

[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. | DOI | MR | JFM

[13] Finocchiaro, C. A.: Prüfer-like conditions on an amalgamated algebra along an ideal. Houston J. Math. 40 (2014), 63-79. | MR | JFM

[14] Fuchs, L.: Über die Ideale arithmetischer Ringe. Comment. Math. Helv. 23 (1949), 334-341 German. | DOI | MR | JFM

[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. | DOI | MR | JFM

[16] Glaz, S.: The weak global dimensions of Gaussian rings. Proc. Am. Math. Soc. 133 (2005), 2507-2513. | DOI | MR | JFM

[17] Griffin, M.: Prüfer rings with zero divisors. J. Reine Angew. Math. 239-240 (1969), 55-67. | DOI | MR | JFM

[18] Kabbaj, S., Louartiti, K., Tamekkante, M.: Bi-amalgamated algebras along ideals. J. Commut. Algebra 9 (2017), 65-87. | DOI | MR | JFM

[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. | DOI | MR | JFM

[20] Koehler, A.: Rings for which every cyclic module is quasi-projective. Math. Ann. 189 (1970), 311-316. | DOI | MR | JFM

[21] Krull, W.: Beiträge zur Arithmetik kommutativer Integritätsbereiche. Math. Z. 41 (1936), 545-577 German. | DOI | MR | JFM

[22] Loper, K. A., Roitman, M.: The content of a Gaussian polynomial is invertible. Proc. Am. Math. Soc. 133 (2005), 1267-1271. | DOI | MR | JFM

[23] Lucas, T. G.: Gaussian polynomials and invertibility. Proc. Am. Math. Soc. 133 (2005), 1881-1886. | DOI | MR | JFM

[24] Prüfer, H.: Untersuchungen über Teilbarkeitseigenschaften in Körpern. J. Reine Angew. Math. 168 (1932), 1-36 German. | DOI | MR | JFM

[25] Tsang, H.: Gauss's Lemma, Ph.D. Thesis. University of Chicago, Chicago (1965). | MR

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