• Published : 2012.01.31

#### Abstract

Let R be a graded Noetherian domain and A a graded Krull overring of R. We show that if h-dim $R\leq2$, then A is a graded Noetherian domain with h-dim $A\leq2$. This is a generalization of the well-know theorem that a Krull overring of a Noetherian domain with dimension $\leq2$ is also a Noetherian domain with dimension $\leq2$.

#### References

1. D. D. Anderson and D. F. Anderson, Divisibility properties of graded domains, Canad. J. Math. 34 (1982), no. 1, 196-215. https://doi.org/10.4153/CJM-1982-013-3
2. R. Gilmer, Commutative Semigroup Rings, University of Chicago Press, Chicago, 1984.
3. W. Heinzer, On Krull overrings of a Noetherian domain, Proc. Amer. Math. Soc. 22 (1969), 217-222. https://doi.org/10.1090/S0002-9939-1969-0254022-7
4. I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
5. H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, Cambridge, 1986.
6. J. Nishimura, Note on integral closures of a Noetherian integral domain, J. Math. Kyoto Univ. 16 (1976), no. 1, 117-122. https://doi.org/10.1215/kjm/1250522963
7. M. H. Park, Integral closure of graded integral domains, Comm. Algebra 35 (2007), no. 12, 3965-3978. https://doi.org/10.1080/00927870701509511
8. C. H. Park and M. H. Park, Integral closure of a graded Noetherian domain, J. Korean Math. Soc. 48 (2011), no. 3, 449-464. https://doi.org/10.4134/JKMS.2011.48.3.449
9. D. E. Rush, Noetherian properties in monoid rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 259-278. https://doi.org/10.1016/S0022-4049(03)00103-8

#### Cited by

1. Graded integral domains in which each nonzero homogeneous t-ideal is divisorial pp.1793-6829, 2018, https://doi.org/10.1142/S021949881950018X
2. Graded-Noetherian property in pullbacks of graded integral domains vol.67, pp.2, 2018, https://doi.org/10.1007/s11587-018-0357-0