Korean J. Math. Vol. 25 No. 2 (2017) pp.215-227
DOI: https://doi.org/10.11568/kjm.2017.25.2.215

Graded integral domains and Nagata rings, II

Main Article Content

Gyu Whan Chang

Abstract

Let $D$ be an integral domain with quotient field $K$, $X$ be an indeterminate over $D$, $K[X]$ be the polynomial ring over $K$, and $R= \{f \in K[X] \mid f(0) \in D\}$; so $R$ is a subring of $K[X]$ containing $D[X]$. For $f = a_0 + a_1X + \cdots + a_nX^n \in R$, let $C(f)$ be the ideal of $R$ generated by $a_0, a_1X, \dots , a_nX^n$ and $N(H) = \{g \in R \mid C(g)_v = R\}$. In this paper, we study two rings $R_{N(H)}$ and Kr$(R, v) = \{\frac{f}{g} \mid f, g \in R$, $g \neq 0$, and $C(f) \subseteq C(g)_v\}$. We then use these two rings to give some examples which show that the results of \cite{ac13} are the best generalizations of Nagata rings and Kronecker function rings to graded integral domains.



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References

[1] D.D. Anderson, Some remarks on the ring R(X), Comment. Math. Univ. St. Paul. 26 (1977), 137–140. Google Scholar

[2] D.D. Anderson and D.F. Anderson, Divisorial ideals and invertible ideals in a graded integral domain, J. Algebra 76 (1982), 549–569. Google Scholar

[3] D.F. Anderson and G.W. Chang, Homogeneous splitting sets of a graded integral domain, J. Algebra 288 (2005), 527-544. Google Scholar

[4] D.F. Anderson and G.W. Chang, Graded integral domains and Nagata rings, J. Algebra 387 (2013), 169–184. Google Scholar

[5] J.T. Arnold, On the ideal theory of the Kronecker function ring and the domain D(X), Canad. J. Math. 21 (1969), 558–563. Google Scholar

[6] G.W. Chang, Pru ̈fer ∗-multiplication domains, Nagata rings, and Kronecker function rings, J. Algebra 319 (2008), 309–319. Google Scholar

[7] L.G. Chouinard II, Krull semigroups and divisor class groups, Canad. J. Math. 33 (1981), 1459–1468. Google Scholar

[8] D. Costa, J. Mott, and M. Zafrullah, The construction D+XDS[X], J. Algebra 53 (1978), 423–439. Google Scholar

[9] F. Decruenaere and E. Jespers, Pru ̈fer domains and graded rings, J. Algebra 53 (1992), 308–320. Google Scholar

[10] M. Fontana and K.A. Loper, A historical overview of Kronecker function rings, Nagata rings, and related star and semistar operations, in: J.W. Brewer, S. Glaz, W.J. Heinzer, B.M. Olberding (Eds.), Multiplicative Ideal Theory in Commu- tative Algebra. A Tribute to the Work of Robert Gilmer, Springer, 2006, pp. 169–187. Google Scholar

[11] R. Gilmer, An embedding theorem for HCF-rings, Proc. Cambridge Philos. Soc. 68 (1970), 583–587. Google Scholar

[12] R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972. Google Scholar

[13] B.G. Kang, Pru ̈fer v-multiplication domains and the ring R[X]Nv , J. Algebra 123 (1989), 151–170. Google Scholar

[14] [16] P. Sahandi, Characterizations of graded Pru ̈fer ∗-multiplication domain, Korean J. Math. 22 (2014), 181–206. Google Scholar

[15] [1] D.D. Anderson, Some remarks on the ring R(X), Comment. Math. Univ. St. Paul. 26 (1977), 137–140. Google Scholar