Torsion pairs and related modules over trivial ring extensions

Year 2024, , 1291 - 1304, 15.10.2024

Lixin Mao

https://doi.org/10.15672/hujms.1272122

Abstract

Let $R\ltimes M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$. We first study how to construct torsion pairs over $R\ltimes M$ from torsion pairs over $R$. Some characterizations of finitely generated (presented) modules, flat modules and coherent rings relative to a torsion pair over $R\ltimes M$ are obtained. Then we discuss the transfers of torsion pairs over $R\ltimes M$ to $R$. Finally, some applications are given in Morita context rings.

Keywords

torsion pair, trivial ring extension, Morita context ring

Supporting Institution

NSFC

Project Number

12171230, 12271249

References

• [1] H. Bass, The Morita Theorems, Mimeographed Notes. University of Oregon, 1962.
• [2] S.E. Dickson, A torsion theory for abelian categories, Trans. Amer. Math. Soc. 121, 223-235, 1966.
• [3] N. Ding and J. Chen, Relative coherence and preenvelopes, Manuscripta Math. 81, 243-262, 1993.
• [4] T. Dumitrescu, N. Mahdou and Y. Zahir, Radical factorization for trivial extensions and amalgamated duplication rings, J. Algebra Appl. 20, 2150025, 2021.
• [5] C. Fan and H. Yao, Torsion pairs in triangulated categories, Adv. Pure Math. 3, 374-379, 2013.
• [6] R.M. Fossum, P. Griffith and I. Reiten, Trivial Extensions of Abelian Categories, Homological Algebra of Trivial Extensions of Abelian Categories with Applications to Ring Theory. Lect. Notes in Math. 456, Springer-Verlag, Berlin, 1975.
• [7] R. Göbel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules, GEM 41, Walter de Gruyter, Berlin-New York, 2006.
• [8] E.L. Green, On the presentation theory of rings in matrix form, Pacific J. Math. 100, 123-138, 1982.
• [9] P. Krylov and A. Tuganbaev, Formal Matrices, Springer International Publishing, Switzerland, 2017.
• [10] X. Ma and Z. Huang, Torsion pairs in recollements of abelian categories, Front. Math. China 13, 875-892, 2018.
• [11] L. Mao, Silting and cosilting modules over trivial ring extensions, Comm. Algebra 51, 1532-1550, 2023.
• [12] L. Mao, Homological properties of trivial ring extensions, J. Algebra Appl. 22, 2350265, 2023.
• [13] K. Morita, Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Diagaku Sect. A6, 83-142, 1958.
• [14] M. Nagata, Local Rings, Interscience Tracts in Math. 13, Interscience New York, 1962.
• [15] I. Palmér and J.E. Roos, Explicit formulae for the global homological dimensions of trivial extensions of rings, J. Algebra 27, 380-413, 1973.
• [16] Y. Peng, X. Ma and Z. Huang, $\tau$-Tilting modules over triangular matrix artin algebras, Internat. J. Algebra Comput. 31, 639-661, 2021.
• [17] I. Reiten, Trivial extensions and Gorenstein rings, Thesis, University of Illinois, Urbana, 1971.
• [18] B. Stenström, Rings of Quotients, Springer, Berlin-Heidelberg-New York, 1975.