On t-almost Dedekind graded domains
1
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
Let $\Gamma$ be a nonzero torsionless commutative cancellative \linebreak monoid with quotient group $\langle \Gamma \rangle$, $R = \bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be a graded integral domain graded by $\Gamma$ such that $R_{\alpha} \neq \{0\}$ for all $\alpha \in \Gamma$, $H$ be the set of nonzero homogeneous elements of $R$, $C(f)$ be the ideal of $R$ generated by the homogeneous components of $f \in R$, and $N(H) = \{f \in R \mid C(f)_v = R\}$. In this paper, we introduce the notion of graded $t$-almost Dedekind domains. We then show that $R$ is a $t$-almost Dedekind domain if and only if $R$ is a graded $t$-almost Dedekind domain and $R_H$ is a $t$-almost Dedekind domains. We also show that if $R = D[\Gamma]$ is the monoid domain of $\Gamma$ over an integral domain $D$, then $R$ is a graded $t$-almost Dedekind domain if and only if $D$ and $\Gamma$ are $t$-almost Dedekind, if and only if $R_{N(H)}$ is an almost Dedekind domain. In particular, if $\langle \Gamma \rangle$ satisfies the ascending chain condition on its cyclic subgroups, then $R = D[\Gamma]$ is a $t$-almost Dedekind domain if and only if $R$ is a graded $t$-almost Dedekind domain.
Abstract
Let $\Gamma$ be a nonzero torsionless commutative cancellative \linebreak monoid with quotient group $\langle \Gamma \rangle$, $R = \bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be a graded integral domain graded by $\Gamma$ such that $R_{\alpha} \neq \{0\}$ for all $\alpha \in \Gamma$, $H$ be the set of nonzero homogeneous elements of $R$, $C(f)$ be the ideal of $R$ generated by the homogeneous components of $f \in R$, and $N(H) = \{f \in R \mid C(f)_v = R\}$. In this paper, we introduce the notion of graded $t$-almost Dedekind domains. We then show that $R$ is a $t$-almost Dedekind domain if and only if $R$ is a graded $t$-almost Dedekind domain and $R_H$ is a $t$-almost Dedekind domains. We also show that if $R = D[\Gamma]$ is the monoid domain of $\Gamma$ over an integral domain $D$, then $R$ is a graded $t$-almost Dedekind domain if and only if $D$ and $\Gamma$ are $t$-almost Dedekind, if and only if $R_{N(H)}$ is an almost Dedekind domain. In particular, if $\langle \Gamma \rangle$ satisfies the ascending chain condition on its cyclic subgroups, then $R = D[\Gamma]$ is a $t$-almost Dedekind domain if and only if $R$ is a graded $t$-almost Dedekind domain.
참고문헌 (17)
- D. D. Anderson, "Divisorial ideals and invertible ideals in a graded integral domain", J. Algebra, 76(2), pp.225~245 (1982)
- D. F. Anderson, "Graded Krull domains", Comm. Algebra, 7(1), pp.79~106 (1979)
- D. F. Anderson, "Homogeneous splitting sets of a graded integral domain", J. Algebra, 288(2), pp.527~544 (2005)
- D. F. Anderson, "Graded integral domains and Nagata rings", J. Algebra, 387, pp.169~184 (2013)
- D. F. Anderson, "Graded Prufer domains", Comm. Algebra
- G. W. Chang, "Characterizations of *-cancellation ideals of an integral domain", Comm. Algebra, 37(9), pp.3309~3320 (2009)
- "J. Algebra", 2010, 3124~3133
- G. W. Chang, "Discrete valuation overrings of a graded Noetherian domain", J. Commutative Algebra
- S. El Bagdahi, "Semistar Dedekind domains", J. Pure Appl. Algebra, 193(1-3), pp.27~60 (2004)
- S. El Baghdadi, "On the class group of a graded domain", J. Pure Appl. Algebra, 171(2-3), pp.171~184 (2002)
- R. Gilmer, "Commutative Semigroup Rings, Chicago Lectures in Mathematics" (1984), Univ. Chicago Press
- R. Gilmer, "Multiplicative Ideal Theory", Queen's papers in Pure and Appl. Math., 90 (1992)
- M. Griffin, "Some results on v-multiplication rings", Canad. J. Math, 97, pp.710~722 (1967)
- F. Halter-Koch, "Ideal Systems. An Introduction to Multiplicative Ideal Theory" (1998), Marcel Dekker
- Nv, "J. Algebra", 1989, 151~170
- K. A. Loper, "Multiplicative Ideal Theory in Commutative Algebra", pp.279~292 (2006), Springer
- D. G. Northcott, "Lessons on Rings, Modules, and Multiplicities" (1968), Cambridge Univ. Press