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Let $R$ be any commutative ring and $S$ be any multiplicative closed set. We introduce an $S$-version of $\mathcal{F}$-Mittag-Leffler modules, called $\mathcal{F_{S}}$-Mittag-Leffler modules, and define the projective dimension with respect to these modules. We give some characterizations of $\mathcal{F_{S}}$-Mittag-Leffler modules, investigate the relationships between $\mathcal{F}$-Mittag-Leffler modules and $\mathcal{F_{S}}$-Mittag-Leffler modules, and use these relations to describe noetherian rings and coherent rings, such as $R$ is noetherian if and only if $R_{S}$ is noetherian and every $\mathcal{F_{S}}$-Mittag-Leffler module is $\mathcal{F}$-Mittag-Leffler. Besides, we also investigate the $\mathcal{M}^\mathcal{F_{S}}$-global dimension of $R$, and prove that $R_{S}$ is noetherian if and only if its $\mathcal{M}^\mathcal{F_{S}}$-global dimension is zero; $R_{S}$ is coherent if and only if its $\mathcal{M}^\mathcal{F_{S}}$-global dimension is at most one.
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참고문헌 (21)
참고문헌 신청
D. D. Anderson / 2006 / S-NOETHERIAN RINGS / Communications in Algebra 30(9):4407~4416
Lidia Angeleri Hugel / 2008 / Mittat-Leffler conditions on modules / Indiana University Mathematics Journal 57(5):2459~2518
Goro Azumaya / 1987 / Finite splitness and finite projectivity / Journal of Algebra 106(1):114~134
Jongwook Baeck / 2016 / $S$-Noetherian Rings and Their Extensions / Taiwanese Journal of Mathematics 20(6):1231~1250
S. Bazzoni / 2018 / S-almost perfect commutative rings / eprint arXiv:1801.04820
Lidia Angeleri Hugel / 2008 / Mittat-Leffler conditions on modules / Indiana University Mathematics Journal 57(5):2459~2518
Goro Azumaya / 1987 / Finite splitness and finite projectivity / Journal of Algebra 106(1):114~134
Jongwook Baeck / 2016 / $S$-Noetherian Rings and Their Extensions / Taiwanese Journal of Mathematics 20(6):1231~1250
S. Bazzoni / 2018 / S-almost perfect commutative rings / eprint arXiv:1801.04820
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