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Year 2024 ,
, 134 - 156, 12.07.2024
Zhanmin Zhu
References
H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press,
Princeton, NJ, 1956.
S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc., 97 (1960),
457-473.
T. J. Cheatham and D. R. Stone, Flat and projective character modules, Proc.
Amer. Math. Soc., 81 (1981), 175-177.
J. L. Chen and N. Q. Ding, A note on existence of envelopes and covers, Bull.
Austral. Math. Soc., 54 (1996), 383-390.
J. L. Chen and N. Q. Ding, On n-coherent rings, Comm. Algebra, 24 (1996),
3211-3216.
N. Q. Ding, Y. L. Li and L. X. Mao, J-coherent rings, J. Algebra Appl., 8
(2009), 139-155.
D. D. Dobbs, On n-flat modules over a commutative ring, Bull. Austral. Math.
Soc., 43 (1991), 491-498.
E. E. Enochs, A note on absolutely pure modules, Canad. Math. Bull., 19
(1976), 361-362.
E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de
Gruyter & Co., Berlin, 2000.
H. Holm and P. Jørgensen, Covers, precovers, and purity, Illinois. J. Math., 52
(2008), 691-703.
W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge University
Press, Cambridge, 2003.
J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New
York-London, 1979.
A. Shamsuddin, n-injective and n-flat modules, Comm. Algebra, 29 (2001),
2039-2050.
B. Stenström, Coherent rings and FP-injective modules, J. London. Math. Soc.,
2 (1970), 323-329.
B. Stenstr¨om, Rings of Quotients, Springer-Verlag, New York-Heidelberg,
1975.
R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach
Science Publishers, Philadelphia, PA, 1991.
X. X. Zhang, J. L. Chen and J. Zhang, On (m, n)-injective modules and (m, n)-
coherent rings, Algebra Colloq., 12 (2005), 149-160.
X. X. Zhang and J. L. Chen, On n-semihereditary and n-coherent rings, Int.
Electron. J. Algebra, 1 (2007), 1-10.
Z. M. Zhu and Z. S. Tan, On n-semihereditary rings, Sci. Math. Jpn., 62 (2005),
455-459.
Z. M. Zhu, I-n-coherent rings, I-n-semihereditary rings, and I-regular rings,
Ukrainian Math. J., 66 (2014), 857-883.
Z. M. Zhu, I-pure submodules, I-FP-injective modules and I-flat modules, Br.
J. Math. Comput. Sci., 8 (2015), 170-188.
Z. M. Zhu, Strongly n-coherent rings, Chinese Ann. Math. Ser. A, 38 (2017),
313-326.
Strongly J-n-Coherent rings
Year 2024 ,
, 134 - 156, 12.07.2024
Zhanmin Zhu
Abstract
Let $R$ be a ring and $n$ a fixed positive integer. A right $R$-module $M$ is called strongly $J$-$n$-injective if every $R$-homomorphism from an $n$-generated small submodule of a free right $R$-module $F$ to $M$ extends to a homomorphism of $F$ to $M$; a right $R$-module $V$ is said to be
strongly $J$-$n$-flat, if for every $n$-generated small submodule $T$ of a free left $R$-module $F$, the canonical map $V\otimes T\rightarrow V\otimes F$ is monic; a ring $R$ is called left strongly $J$-$n$-coherent if every $n$-generated small submodule of a free left $R$-module is finitely presented; a ring $R$ is said to be left $J$-$n$-semihereditary if every $n$-generated small left ideal of $R$ is projective. We study strongly $J$-$n$-injective modules, strongly $J$-$n$-flat modules and left strongly $J$-$n$-coherent rings. Using the concepts of strongly $J$-$n$-injectivity and strongly $J$-$n$-flatness of modules, we also present some characterizations of strongly $J$-$n$-coherent rings and $J$-$n$-semihereditary rings.
References
H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press,
Princeton, NJ, 1956.
S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc., 97 (1960),
457-473.
T. J. Cheatham and D. R. Stone, Flat and projective character modules, Proc.
Amer. Math. Soc., 81 (1981), 175-177.
J. L. Chen and N. Q. Ding, A note on existence of envelopes and covers, Bull.
Austral. Math. Soc., 54 (1996), 383-390.
J. L. Chen and N. Q. Ding, On n-coherent rings, Comm. Algebra, 24 (1996),
3211-3216.
N. Q. Ding, Y. L. Li and L. X. Mao, J-coherent rings, J. Algebra Appl., 8
(2009), 139-155.
D. D. Dobbs, On n-flat modules over a commutative ring, Bull. Austral. Math.
Soc., 43 (1991), 491-498.
E. E. Enochs, A note on absolutely pure modules, Canad. Math. Bull., 19
(1976), 361-362.
E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de
Gruyter & Co., Berlin, 2000.
H. Holm and P. Jørgensen, Covers, precovers, and purity, Illinois. J. Math., 52
(2008), 691-703.
W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge University
Press, Cambridge, 2003.
J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New
York-London, 1979.
A. Shamsuddin, n-injective and n-flat modules, Comm. Algebra, 29 (2001),
2039-2050.
B. Stenström, Coherent rings and FP-injective modules, J. London. Math. Soc.,
2 (1970), 323-329.
B. Stenstr¨om, Rings of Quotients, Springer-Verlag, New York-Heidelberg,
1975.
R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach
Science Publishers, Philadelphia, PA, 1991.
X. X. Zhang, J. L. Chen and J. Zhang, On (m, n)-injective modules and (m, n)-
coherent rings, Algebra Colloq., 12 (2005), 149-160.
X. X. Zhang and J. L. Chen, On n-semihereditary and n-coherent rings, Int.
Electron. J. Algebra, 1 (2007), 1-10.
Z. M. Zhu and Z. S. Tan, On n-semihereditary rings, Sci. Math. Jpn., 62 (2005),
455-459.
Z. M. Zhu, I-n-coherent rings, I-n-semihereditary rings, and I-regular rings,
Ukrainian Math. J., 66 (2014), 857-883.
Z. M. Zhu, I-pure submodules, I-FP-injective modules and I-flat modules, Br.
J. Math. Comput. Sci., 8 (2015), 170-188.
Z. M. Zhu, Strongly n-coherent rings, Chinese Ann. Math. Ser. A, 38 (2017),
313-326.
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There are 22 citations in total.
Details
Primary Language
English
Subjects
Algebra and Number Theory
Journal Section
Articles
Authors
Zhanmin Zhu
This is me
Jiaxing University
China
Early Pub Date
December 28, 2023
Publication Date
July 12, 2024
Published in Issue
Year 2024
Cite
APA
Zhu, Z. (2024). Strongly J-n-Coherent rings. International Electronic Journal of Algebra, 36(36), 134-156. https://doi.org/10.24330/ieja.1411161
AMA
Zhu Z. Strongly J-n-Coherent rings. IEJA. July 2024;36(36):134-156. doi:10.24330/ieja.1411161
Chicago
Zhu, Zhanmin. “Strongly J-N-Coherent Rings”. International Electronic Journal of Algebra 36, no. 36 (July 2024): 134-56. https://doi.org/10.24330/ieja.1411161.
EndNote
Zhu Z (July 1, 2024) Strongly J-n-Coherent rings. International Electronic Journal of Algebra 36 36 134–156.
IEEE
Z. Zhu, “Strongly J-n-Coherent rings”, IEJA , vol. 36, no. 36, pp. 134–156, 2024, doi: 10.24330/ieja.1411161.
ISNAD
Zhu, Zhanmin. “Strongly J-N-Coherent Rings”. International Electronic Journal of Algebra 36/36 (July 2024), 134-156. https://doi.org/10.24330/ieja.1411161.
JAMA
Zhu Z. Strongly J-n-Coherent rings. IEJA . 2024;36:134–156.
MLA
Zhu, Zhanmin. “Strongly J-N-Coherent Rings”. International Electronic Journal of Algebra, vol. 36, no. 36, 2024, pp. 134-56, doi:10.24330/ieja.1411161.
Vancouver
Zhu Z. Strongly J-n-Coherent rings. IEJA. 2024;36(36):134-56.