Strongly J-n-Coherent rings

Year 2024, Early Access, 1 - 23

Zhanmin Zhu

https://doi.org/10.24330/ieja.1411161

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.

Keywords

Strongly $J$-$n$-injective module, strongly $J$-$n$-flat module, strongly $J$-$n$-coherent ring, $J$-$n$-semihereditary ring

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.