Graded integral domains in which each nonzero homogeneous ideal is divisorial

다국어 초록 (Multilingual Abstract)

Let $\Gamma$ be a nonzero commutative cancellative monoid (written additively), $R = \bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a $\Gamma$-graded integral domain with $R_{\alpha} \neq \{0\}$ for all $\alpha \in \Gamma$, and $S(H) = \{f \in R \,|\, C(f) = R\}$. In this paper, we study homogeneously divisorial domains which are graded integral domains whose nonzero homogeneous ideals are divisorial. Among other things, we show that if $R$ is integrally closed, then $R$ is a homogeneously divisorial domain if and only if $R_{S(H)}$ is an h-local Pr\"ufer domain whose maximal ideals are invertible, if and only if $R$ satisfies the following four conditions: (i) $R$ is a graded-Pr\"{u}fer domain, (ii) every homogeneous maximal ideal of $R$ is invertible, (iii) each nonzero homogeneous prime ideal of $R$ is contained in a unique homogeneous maximal ideal, and (iv) each homogeneous ideal of $R$ has only finitely many minimal prime ideals. We also show that if $R$ is a graded-Noetherian domain, then $R$ is a homogeneously divisorial domain if and only if $R_{S(H)}$ is a divisorial domain of (Krull) dimension one.

참고문헌 (Reference)

1 S. Bazzoni, "Warfield Domains" Elsevier BV 185 (185): 836-868, 1996

2 Eben Matlis, "Reflexive domains" Elsevier BV 8 (8): 1-33, 1968

3 B.G Kang, "Prüfer v-multiplication domains and the ring R[X]Nv" Elsevier BV 123 (123): 151-170, 1989

4 Muhammad Zafrullah, "On finite conductor domains" Springer Science and Business Media LLC 24 (24): 191-204, 1978

5 David E. Rush, "Noetherian properties in monoid rings" Elsevier BV 185 (185): 259-278, 2003

6 R. Gilmer, "Multiplicative Ideal Theory" Marcel Dekker, Inc. 1972

7 D. G. Northcott, "Lessons on Rings, Modules and Multiplicities" Cambridge University Press 1968

8 William Heinzer, "Integral domains in which each non‐zero ideal is divisorial" Wiley 15 (15): 164-170, 1968

9 Mi Hee Park, "Integral Closure of Graded Integral Domains" Informa UK Limited 35 (35): 3965-3978, 2007

10 박창환, "INTEGRAL CLOSURE OF A GRADED NOETHERIAN DOMAIN" 대한수학회 48 (48): 449-464, 2011

11 David F. Anderson, "Graded integral domains whose nonzero homogeneous ideals are invertible" World Scientific Pub Co Pte Lt 26 (26): 1361-1368, 2016

12 David F. Anderson, "Graded integral domains and Nagata rings" Elsevier BV 387 : 169-184, 2013

13 David F. Anderson, "Graded Prüfer domains" Informa UK Limited 46 (46): 792-809, 2017

14 D. D. Anderson, "Divisibility Properties of Graded Domains" Canadian Mathematical Society 34 (34): 196-215, 1982

15 E. Matlis, "Cotorsion modules" 49 : 66-, 1964

16 M. Kreuzer, "Computational Commutative Algebra. 2" Springer-Verlag 2005

17 Parviz Sahandi, "Characterizations of graded Pru"fer star-multiplication domains" 강원경기수학회 22 (22): 181-206, 2014

18 Parviz Sahandi, "Characterizations of Graded Prüfer $$\star $$ ⋆ -Multiplication Domains, II" Springer Science and Business Media LLC 44 (44): 61-78, 2018