# SOME REMARKS ON TYPES OF NOETHERIAN LOCAL RINGS

• Lee, Kisuk (Department of Mathematics Sookmyung Women's University)
• Accepted : 2014.10.10
• Published : 2014.11.15

#### Abstract

We study some results which concern the types of Noetherian local rings, and improve slightly the previous result: For a complete unmixed (or quasi-unmixed) Noetherian local ring A, we prove that if either $A_p$ is Cohen-Macaulay, or $r(Ap){\leq}depth$ $A_p+1$ for every prime ideal p in A, then A is Cohen-Macaulay. Also, some analogous results for modules are considered.

#### References

1. Y. Aoyama, Complete local ($S_{n-1}$) rings of type n$\geq$3 are Cohen-Macaulay, Proc. Japan Acard. 70 Ser. A (1994), 80-83. https://doi.org/10.3792/pjaa.70.80
2. H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28. https://doi.org/10.1007/BF01112819
3. W. Bruns, The Evans- Griffith syzygy theorem and Bass numbers, Proc. Amer. Math. Soc. 115 (1992), 939-946. https://doi.org/10.1090/S0002-9939-1992-1088439-0
4. D. Costa, C. Huneke, and M. Miller, Complete local domains of type two are Cohen-Macaulay, Bull. London Math. Soc. 17 (1985), 29-31. https://doi.org/10.1112/blms/17.1.29
5. H. B. Foxby, On the i in a minimal injective resolution II, Math. Scand. 41 (1977), 19-44. https://doi.org/10.7146/math.scand.a-11700
6. T. Kawasaki, Local rings of relatively small type are Cohen-Macaulay, Proc. Amer. Math. Soc. 122 (1994), 703-709. https://doi.org/10.1090/S0002-9939-1994-1215029-0
7. K. Lee, A note on types of Noetherian local rings, Bull. Korean Math. Soc. 39 (2002), no. 4, 645-652. https://doi.org/10.4134/BKMS.2002.39.4.645
8. K. Lee, On types of Noetherian local rings and modules, J. Korean Math. Soc. 44 (2007), no. 4, 987-995. https://doi.org/10.4134/JKMS.2007.44.4.987
9. K. Lee, Maps in minimal injective resolutions of modules, Bull. Korean Math. Soc. 46 (2009), no. 3, 545-551. https://doi.org/10.4134/BKMS.2009.46.3.545
10. T. Marley, Unmixed local rings of type two are Cohen-Macaulay, Bull. London Math. Soc. 23 (1991), 43-45. https://doi.org/10.1112/blms/23.1.43
11. H. Matsumura, Commutative ring theory, Camb. Study Adv. Math. 8, Cambridge 1986.
12. P. Roberts, Homological invariants of modules over commutative rings, Sem. Math. Sup., Presses Univ. Montreal, Montreal (1980).
13. P. Roberts, Rings of type 1 are Gorenstein, Bull. London Math. Soc. 15 (1983), 48-50. https://doi.org/10.1112/blms/15.1.48
14. W. V. Vasconcelos, Divisor theory in module categories, Math. Studies 14, North Holland Publ. Co., Amsterdam, 1975.