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The amalgamated duplication of a ring along a semidualizing ideal. (English) Zbl 1279.13025
Let \(R\) be a commutative Noetherian ring and let \(I\) be an ideal of \(R\). In this paper, after recalling briefly the main properties of the amalgamated duplication ring \((R\bowtie I)\) which is introduced by M. D’Anna and M. Fontana [J. Algebra Appl. 6, No. 3, 443–459 (2007; Zbl 1126.13002)], we restrict our attention to the study of the properties of \((R\bowtie I)\), when \(I\) is a semidualizing ideal of \(R\), i.e., \(I\) is an ideal of \(R\) and \(I\) is a semidualizing \(R\)-module. In particular, it is shown that if \(I\) is a semidualizing ideal and \(M\) is a finitely generated \(R\)-module, then \(M\) is totally \(I\)-reflexive as an \(R\)-module if and only if \(M\) is totally reflexive as an \((R\bowtie I)\)-module. In addition, it is shown that if \(I\) is a semidualizing ideal, then \(R\) and \(I\) are Gorenstein projective over \((R\bowtie I)\), and every injective R-module is Gorenstein injective as an \((R\bowtie I)\)-module. Finally, it is proved that if \(I\) is a non-zero flat ideal of \(R\), then \(\text{fd}_{R}(M) = \text{fd} _{R \bowtie I} (M \otimes _{R} (R\bowtie I)) = \text{fd} _{R} (M\otimes _{R}(R\bowtie I))\), for every \(R\)-module \(M\).

MSC:
13D05 Homological dimension and commutative rings
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[1] A. BAGHERI - M. SALIMI - E. TAVASOLI - S. YASSEMI, A Construction of Quasi- Gorenstein Rings, J. Algebra Appl. 11, no. 1 (2012), 1250013 (9 pages). · Zbl 1245.13017 · doi:10.1142/S0219498811005361
[2] W. BRUNS - J. HERZOG, Cohen-Macaulay Rings, Cambridge University press, Cambridge, 1993.
[3] L. W. CHRISTENSEN, Gorenstein Dimensions, Lecture Notes in Math. Vol. 1747, Springer, Berlin, 2000. 127
[4] L. W. CHRISTENSEN - H. B. FOXBY - H. HOLM, Beyond Totally Reflexive Modules and Back: a survey on Gorenstein dimensions, “Commutative Algebra-Noetherian and non-Noetherian Perspectives”, 101-143, Springer, New York, 2011. · Zbl 1225.13019 · doi:10.1007/978-1-4419-6990-3_5
[5] M. D’ANNA, A construction of Gorenstein rings, J. Algebra 306 (2006), pp. 507-519. · Zbl 1120.13022 · doi:10.1016/j.jalgebra.2005.12.023
[6] M. D’ANNA - M. FONTANA, An amalgamated duplication of a ring along an ideal, J. Algebra Appl. 6, no. 3 (2007), pp. 443-459. · Zbl 1126.13002 · doi:10.1142/S0219498807002326 · arxiv:math/0605602
[7] E. E. ENOCHS - O. M. G. JENDA, Relative Homological Algebra, de Gruyter Expositions in Mathematics, vol. 30, Walter de Gruyter and Co., Berlin, 2000.
[8] N. EPSTEIN - Y. YAO, Criteria for Flatness and Injectivity, Math. Z. 271, no. 3-4 (2012), pp. 1193-1210.
[9] H. B. FOXBY, Gorenstein Modules and Related Modules, Math. Scand. 31, (1972), pp. 267-284. · Zbl 0272.13009 · eudml:166275
[10] E. S. GOLOD, G-dimension and Generalized Perfect Ideals, Trudy Mat. Inst. Steklov. 165, (1984), pp. 62-66.
[11] H. HOLM - P. JéRGENSEN, Cohen-Macaulay Homological Dimensions, Rend. Semin. Mat. Univ. Padova, 117 (2007), pp. 87-112.
[12] M. CHHITI - N. MAHDOU, Homological dimension of the amalgamated duplication of a ring along a pure ideal, Afr. Diaspora J. Math. (N. S.) 10, no. 1, (2010), pp. 1-6. · Zbl 1238.13022 · euclid:adjm/1274101579 · arxiv:0910.0644
[13] N. MAHDOU - M. TAMEKKANTE, Gorenstein global dimension of an amalga- mated duplication of a coherent ring along an ideal, Mediterr. J. Math. 8, no. 3 (2011), pp. 293-305. · Zbl 1229.18013 · doi:10.1007/s00009-010-0090-8 · arxiv:0909.4863
[14] M. NAGATA, Local Rings , Interscience, New York, 1962.
[15] S. SATHER-WAGSTAFF, Semidualizing modules, URL:http://www.ndsu.edu/ pubweb/\(ssatherw/.\)
[16] J. SHAPIRO, On a construction of Gorenstein rings proposed by M. D’Anna, J. Algebra, 323 (2010), pp. 1155-1158.
[17] E. TAVASOLI - M. SALIMI - A. TEHRANIAN, Amalgamated duplication of some special rings, Bulletin of Korean Math. Soc. 49 no. 5 (2012) pp. 989-996. · Zbl 1255.13015 · doi:10.4134/BKMS.2012.49.5.989
[18] W. V. VASCONCELOS, Divisor Theory in Module Categories, North-Holland Math. Stud., vol. 14, North-Holland Publishing Co., Amsterdam, (1974). · Zbl 0296.13005
[19] T. WAKAMATSU, On modules with trivial self-extensions, J. Algebra, 114, no. 1 (1988), pp. 106-114. · Zbl 0646.16025 · doi:10.1016/0021-8693(88)90215-3
[20] S. YASSEMI, On flat and injective dimension, Italian Journal of Pure and Applied Mathematics, N. 6 (1999), pp. 33-41. · Zbl 0964.16007
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