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Keywords:
amalgamation of ring; amalgamation of module; Cohen-Macaulay; injective module; projective(flat) module
amalgamation of ring; amalgamation of module; Cohen-Macaulay; injective module; projective(flat) module
Summary:
Let $R$ and $S$ be commutative rings with identity, $J$ be an ideal of $S$, $f \colon R \to S$ be a ring homomorphism, $M$ be an $R$-module, $N$ be an $S$-module, and let $\varphi \colon M \to N$ be an $R$-homomorphism. The amalgamation of $R$ with $S$ along $J$ with respect to $f$ denoted by $R \bowtie ^{f} J$ was introduced by M. D'Anna et al. (2010). Recently, R. El Khalfaoui et al. (2021) introduced a special kind of $(R \bowtie ^{f} J)$-module called the amalgamation of $M$ and $N$ along $J$ with respect to $\varphi $, and denoted by $M \bowtie ^{\varphi } JN$. We study some homological properties of the $(R \bowtie ^{f} J)$-module $M \bowtie ^{\varphi } JN$. Among other results, we investigate projectivity, flatness, injectivity, Cohen-Macaulayness, and prime property of the $(R \bowtie ^{f} J)$-module $M \bowtie ^{\varphi } JN$ in connection to their corresponding properties of the $R$-modules $M$ and $JN$.
References:
[1] Bouba, E. M., Mahdou, N., Tamekkante, M.: Duplication of a module along an ideal. Acta Math. Hung. 154 (2018), 29-42. DOI 10.1007/s10474-017-0775-6 | MR 3746520 | Zbl 1399.13011
[2] Brodmann, M. P., Sharp, R. Y.: Local Cohomology: An Algebraic Introduction with Geometric Applications. Cambridge Studies in Advanced Mathematics 60. Cambridge University Press, Cambridge (1998). DOI 10.1017/CBO9780511629204 | MR 1613627 | Zbl 0903.13006
[3] D'Anna, M.: A construction of Gorenstein rings. J. Algebra 306 (2006), 507-519. DOI 10.1016/j.jalgebra.2005.12.023 | MR 2271349 | Zbl 1120.13022
[4] D'Anna, M., Finocchiaro, C. A., Fontana, M.: Amalgamated algebras along an ideal. Commutative Algebra and Its Applications Walter de Gruyter, Berlin (2009), 155-172. DOI 10.1515/9783110213188.155 | MR 2606283 | Zbl 1177.13043
[5] 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 10.1016/j.jpaa.2009.12.008 | MR 2593689 | Zbl 1191.13006
[6] 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 10.1142/S0219498807002326 | MR 2337762 | Zbl 1126.13002
[7] D'Anna, M., Fontana, M.: The amalgamated duplication of a ring along a multiplicative-cannonical ideal. Ark. Mat. 45 (2007), 241-252. DOI 10.1007/s11512-006-0038-1 | MR 2342602 | Zbl 1143.13002
[8] Khalfaoui, R. El, Mahdou, N., Sahandi, P., Shirmohammadi, N.: Amalgamated modules along an ideal. Commun. Korean Math. Soc. 36 (2021), 1-10. DOI 10.4134/CKMS.c200064 | MR 4215837 | Zbl 1467.13026
[9] Enochs, E.: Flat covers and flat cotorsion modules. Proc. Am. Math. Soc. 92 (1984), 179-184. DOI 10.1090/S0002-9939-1984-0754698-X | MR 0754698 | Zbl 0522.13008
[10] Kosmatov, N. V.: Bounds for the homological dimensions of pullbacks. J. Math. Sci., New York 112 (2002), 4367-4370. DOI 10.1023/A:1020351104689 | MR 1757826 | Zbl 1052.16006
[11] Salimi, M., Tavasoli, E., Yassemi, S.: The amalgamated duplication of a ring along a semidualizing ideal. Rend. Semin. Mat. Univ. Padova 129 (2013), 115-127. DOI 10.4171/RSMUP/129-8 | MR 3090634 | Zbl 1279.13025
[12] Shapiro, J.: On a construction of Gorenstein rings proposed by M. D'Anna. J. Algebra 323 (2010), 1155-1158. DOI 10.1016/j.jalgebra.2009.12.003 | MR 2578598 | Zbl 1184.13069
[13] Tavasoli, E.: Some homological properties of amalgamation. Mat. Vesn. 68 (2016), 254-258. MR 3554642 | Zbl 1458.13026
[14] Tiraş, Y., Tercan, A., Harmanci, A.: Prime modules. Honam Math. J. 18 (1996), 5-15. MR 1402357 | Zbl 0948.13004
[15] Xu, J.: Flat Covers of Modules. Lecture Notes in Mathematics 1634. Springer, Berlin (1996). DOI 10.1007/BFb0094173 | MR 1438789 | Zbl 0860.16002
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