The Development of Non-Noetherian Grade and Its Applications
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Hochster and Barger were the first to introduce notions of grade for non-Noetherian rings. Their work laid the foundation for Alfonsi’s work which unified and generalized these earlier definitions. The various notions of grade have played an important role in the development of the theory of coherent rings. This paper looks at the historical development of non-Noetherian grade, as well as its applications.
KeywordsNon-Noetherian ring Coherent ring Grade Depth Polynomial grade
Mathematics Subject Classification (2010):13-02 13C15 13D05
The author would like to thank the referee for their thorough reading and helpful comments to improve this paper. In addition, the author thanks Sarah Glaz for her suggestion regarding the need for the current work, as well as for her assistance and continued encouragement.
- 2.M. Auslander, M. Bridger, Stable Module Theory, Memoirs of the American Mathematical Society, vol. 94 (American Mathematical Society, Providence, R.I., 1969)Google Scholar
- 6.Sarah Glaz, Commutative coherent rings: historical perspective and current developments. Nieuw Arch. Wisk. (4) 10(1–2), 37–56 (1992)Google Scholar
- 8.M. Hochster,Grade-sensitive modules and perfect modules. Proc. London Math. Soc. (3) 29, 55–76 (1974)Google Scholar
- 9.L. Hummel, T. Marley, The Auslander-Bridger formula and the Gorenstein property for coherent rings. J. Comm. Alg. 1 (2009)Google Scholar
- 11.P. Jaffard, Théorie de la Dimension Dans les Anneaux de Polynomes, Mémorial Science Mathematics Fasc. vol. 146 (Gauthier-Villars, Paris, 1960)Google Scholar
- 14.K.P. Mcdowell, Commutative Coherent Rings, ProQuest LLC, Ann Arbor, MI, 1974, Thesis (Ph.D.)–McMaster University (Canada)Google Scholar
- 15.A. Mohsen, M. Tousi, On the notion of Cohen-Macaulayness for non-Noetherian rings. J. Algebra 322, (2009)Google Scholar
- 16.D.G. Northcott, Finite Free Resolutions, Cambridge Tracts in Mathematics, vol. 71 (Cambridge University Press, Cambridge, 1976)Google Scholar
- 17.D.G. Northcott, Projective ideals and MacRae’s invariant. J. London Math. Soc. (2) 24(2), 211–226 (1981)Google Scholar
- 23.J.R. Strooker, Homological Questions in Local Algebra, London Mathematical Society Lecture Note Series, vol. 145 (Cambridge University Press, Cambridge, 1990)Google Scholar