• Honam Mathematical Journal
• /
• v.38 no.2
• /
• pp.231-241
• /
• 2016

In this paper we will define the tight integral closure of a finite set of ideals of a ring relative to a module and we will study some related results.

• Honam Mathematical Journal
• /
• v.19 no.1
• /
• pp.47-52
• /
• 1997

• Honam Mathematical Journal
• /
• v.24 no.1
• /
• pp.1-8
• /
• 2002

In this paper we will show that if E is an injective module over a commutative ring A, then the sequence of sets $Ass_{A}(A/I^{n})^{*(E)}),\;n{\in}N,$ is increasing and ultimately constant. Also we will obtain some results concerning the integral closure of ideals related to some modules.

• Journal of the Korean Mathematical Society
• /
• v.48 no.3
• /
• pp.449-464
• /
• 2011

We show that, if R is a graded Noetherian ring and I is a proper ideal of R generated by n homogeneous elements, then any prime ideal of R minimal over I has h-height ${\leq}$ n, and that if R is a graded Noetherian domain with h-dim R ${\leq}$ 2, then the integral closure R' of R is also a graded Noetherian domain with h-dim R' ${\leq}$ 2. We also present a short improved proof of the result that, if R is a graded Noetherian domain, then the integral closure of R is a graded Krull domain.

• Honam Mathematical Journal
• /
• v.29 no.4
• /
• pp.653-666
• /
• 2007

Let R be a commutative Noetherian ring (with a nonzero identity) and let M be an R-module. Further let I be an ideal of R. In this paper, by putting a suitable condition on $Ass_R$(M), we obtain some results concerning $I^{*(M)}$ and prove that the sequence of sets $Ass_R(R/(I^n)^{*(M)})$, $n\;\in\;N$, is increasing and ultimately constant. (Here $(I^n)^{*(M)}$ denotes the integral closure of $I^n$ relative to M.)

• Bulletin of the Korean Mathematical Society
• /
• v.29 no.1
• /
• pp.65-72
• /
• 1992

All rings are commutative, Noetherian with identity and of prime characteristic p, unless otherwise specified. First, we describe the definition of tight closure of an ideal and the properties about the tight closure used frequently. The technique used here for the tight closure was introduced by M. Hochster and C. Huneke [4,5, or 6]. Using the concepts of the tight closure and its properties, we will prove that if R is a complete local domain and F-rational, then R is Cohen-Macaulay. Next, we study the properties of R$^{+}$, the integral closure of a domain in an algebraic closure of its field of fractions. In fact, if R is a complete local domain of characteristic p>0, then R$^{+}$ is Cohen-Macaulay [8]. But we do not know this fact is true or not if the characteristic of R is zero. For the special case we can show that if R is a non-Cohen-Macaulay normal domain containing the rationals Q, then R$^{+}$ is not Cohen-Macaulay. Finally we will prove that if R is an excellent local domain of characteristic p and F-ratiional, then R is Cohen-Macaulay.aulay.

• Journal of the Korean Mathematical Society
• /
• v.52 no.1
• /
• pp.1-21
• /
• 2015

C-domains are defined via class semigroups, and every C-domain is a Mori domain with nonzero conductor whose complete integral closure is a Krull domain with finite class group. In order to extend the concept of C-domains to rings with zero divisors, we study v-Marot rings as generalizations of ordinary Marot rings and investigate their theory of regular divisorial ideals. Based on this we establish a generalization of a result well-known for integral domains. Let R be a v-Marot Mori ring, $\hat{R}$ its complete integral closure, and suppose that the conductor f = (R : $\hat{R}$) is regular. If the residue class ring R/f and the class group C($\hat{R}$) are both finite, then R is a C-ring. Moreover, we study both v-Marot rings and C-rings under various ring extensions.

• East Asian mathematical journal
• /
• v.10 no.1
• /
• pp.199-202
• /
• 1994

• Bulletin of the Korean Mathematical Society
• /
• v.40 no.3
• /
• pp.509-520
• /
• 2003

Let p be a prime number, $Q_{p}$ the field of p-adic numbers, K a finite extension of $Q_{p}$, $\bar{K}}$ a fixed algebraic closure of K and $C_{p}$ the completion of K with respect to the p-adic valuation. Let E be a closed subfield of $C_{p}$, containing K. Given elements $t_1$...,$t_{r}$ $\in$ E for which the field K($t_1$...,$t_{r}$) is dense in E, we construct integral bases of E over K.

• Bulletin of the Korean Mathematical Society
• /
• v.46 no.1
• /
• pp.61-66
• /
• 2009

We provide a generalized Myers theorem under integral curvature bound and use this result to obtain a closure theorem in general relativity.