|
|
Pure Mathematics 2025
多项式环上的n-Gorenstein投射模
|
Abstract:
设n是一个正整数，R是环，本文讨论了多项式环上的n-Gorenstein投射模和n-Gorenstein内射模的结构。证明了若(Y,f)是n-Gorenstein投射左R[x]-模，则f是单同态，coker f是(n-1)-Gorenstein~投射左R-模；对偶地，(Y,f)是n-Gorenstein内射左R[x]-模，则f是满同态，Ker f是(n-1)-Gorenstein~内射左R-模。
Let n be a positive integer and R be a ring. In this paper,n-Gorenstein projective modules and n-Gorenstein injective modules over polynomial rings were introduced.It is proved that if (Y,f) is an n-Gorenstein projective left R[x]-module, then f is a monomorphism and coker f is (n-1)-Gorenstein projective left R-module. Dually, if (Y,f) is an n-Gorenstein injective left R[x]-module, then f is an epimorphism and Ker f is (n-1)-Gorenstein injective left R-module.
| [1] | Auslander, M. and Bridger, M. (1969) Stable Module Theory. In: Memoirs of the American
Mathematical Society, AMS. https://doi.org/10.1090/memo/0094 |
| [2] | Enochs, E.E. and Jenda, O.M.G. (1995) Gorenstein Injective and Projective Modules. Mathe-
matische Zeitschrift, 220, 611-633. https://doi.org/10.1007/bf02572634 |
| [3] | Tang, X. (2015) Applications of N-Gorenstein Projective and Injective Modules. Hacettepe
Journal of Mathematics and Statistics, 44, 1435-1443.
https://doi.org/10.15672/hjms.2015449673 |
| [4] | Quillen, D. (1976) Projective Modules over Polynomial Rings. Inventiones Mathematicae, 36,
167-171. https://doi.org/10.1007/bf01390008 |
| [5] | Izelgue, L., Mimouni, A.A. and Tamoussit, A. (2019) On the Module Structure of the Integer-
Valued Polynomial Rings. Bulletin of the Malaysian Mathematical Sciences Society, 43, 2687-
2699. https://doi.org/10.1007/s40840-019-00826-5 |
| [6] | Mao, L. (2024) A Structure Result on Modules over Polynomial Rings. Journal of Algebra and
Its Applications, 24, Article 2550230. https://doi.org/10.1142/s0219498825502305 |
| [7] | Mao, L. (2024) Gorenstein Projective, Injective, Flat Modules and Dimensions over Polynomial
Rings. Journal of Algebra and Its Applications. https://doi.org/10.1142/s0219498826500568 |
| [8] | Enochs, E.E., Cortes-Izurdiaga, M. and Torrecillas, B. (2014) Gorenstein Conditions over
Triangular Matrix Rings. Journal of Pure and Applied Algebra, 218, 1544-1554.
https://doi.org/10.1016/j.jpaa.2013.12.006 |
| [9] | Holm, H. (2004) Gorenstein Homological Dimensions. Journal of Pure and Applied Algebra,
189, 167-193. https://doi.org/10.1016/j.jpaa.2003.11.007 |
