A Gorenstein homological characterization of Krull domains
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In this note, we shed new light on Krull domains from the point view of Gorenstein homological algebra. By using the so-called $w$-operation, we show that an integral domain $R$ is Krull if and only if for any nonzero proper $w$-ideal $I$, the Gorenstein global dimension of the $w$-factor ring $(R/I)_{w}$ is zero. Further, we obtain that an integral domain $R$ is Dedekind if and only if for any nonzero proper ideal $I$, the Gorenstein global dimension of the factor ring $R/I$ is zero.
Abstract
In this note, we shed new light on Krull domains from the point view of Gorenstein homological algebra. By using the so-called $w$-operation, we show that an integral domain $R$ is Krull if and only if for any nonzero proper $w$-ideal $I$, the Gorenstein global dimension of the $w$-factor ring $(R/I)_{w}$ is zero. Further, we obtain that an integral domain $R$ is Dedekind if and only if for any nonzero proper ideal $I$, the Gorenstein global dimension of the factor ring $R/I$ is zero.
참고문헌 (21)
- D. D. Anderson, "Finite character representations for integral domains", Boll. Un. Mat. Ital. B, 3, pp.613~630 (1992)
- H. Bass, "Finitistic dimension and a homological generalization of semi-primary rings", Trans. Amer. Math. Soc, 95, pp.466~488 (1960)[DOI]
- D. Bennis, "On 2-SG-semisimple rings", Rocky Mountain J. Math, 45(4), pp.1093~1100 (2015)[DOI]
- D. Bennis, "Strongly Gorenstein projective, injective, and flat modules", J. Pure Appl. Algebra, 210(2), pp.437~445 (2007)[DOI]
- D. Bennis, "A generalization of strongly Gorenstein projective modules", J. Algebra Appl, 8(2), pp.219~227 (2009)[DOI]
- D. Bennis, "Global Gorenstein dimensions", Proc. Amer. Math. Soc, 138(2), pp.461~465 (2010)[DOI]
- D. Bennis, "Rings over which all modules are strongly Gorenstein projective", Rocky Mountain J. Math, 40(3), pp.749~759 (2010)[DOI]
- G. W. Chang, "A characterization of Krull domains in terms of their factor rings", Comm. Algebra, 51(3), pp.1280~1292 (2023)[DOI]
- N. Ding, "The flat dimensions of injective modules", Manuscripta Math, 78(2), pp.165~177 (1993)[DOI]
- E. E. Enochs, "Gorenstein injective and projective modules", Math. Z, 220(4), pp.611~633 (1995)[DOI]
- K. Hu, "A note on Artinian local rings", Bull. Korean Math. Soc, 59(5), pp.1317~1325 (2022)[DOI]
- J. L. Mott, "On Krull domains", Arch. Math. (Basel), 56(6), pp.559~568 (1991)[DOI]
- M. Raynaud, "Crit`eres de platitude et de projectivit´e. Techniques de"platification"d’un module", Invent. Math, 13, pp.1~89 (1971)[DOI]
- F. Wang, "Foundations of commutative rings and their modules" (2016), Springer
- F. Wang, "Finitistic weak dimensions of pullbacks", J. Pure Appl. Algebra, 224(6), pp.106274~ (2020)[DOI]
- F. Wang, "On strong Mori domains", J. Pure Appl. Algebra, 135(2), pp.155~165 (1999)[DOI]
- F. Wang, "Every Pr¨ufer ring does not have small finitistic dimension at most one", Comm. Algebra, 48(12), pp.5311~5320 (2020)[DOI]
- X. Wu, "A generalization of w-linked extensions", Bull. Korean Math. Soc, 59(3), pp.725~743 (2022)[DOI]
- H. Yin, "w-modules over commutative rings", J. Korean Math. Soc, 48(1), pp.207~222 (2011)[DOI]
- X. Zhang, "The small finitistic dimensions of commutative rings", J. Commut. Algebra, 15(1), pp.131~138 (2023)
- D. Zhou, "A Cohen-type theorem for w-Artinian modules", J. Algebra Appl, 20(6), pp.2150106~ (2021)[DOI]