• Honam Mathematical Journal
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• v.21 no.1
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• pp.13-18
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• 1999

• Bulletin of the Korean Mathematical Society
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• v.31 no.2
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• pp.210-214
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• 1994

The purpose of this paper is to investigate the relationship between the depths of the Rees algebra R[It] and the associated graded ring g $r_{I}$(R) of an ideal I in a local ring (R,m) of dim(R) > 0. The relationship between the Cohen-Macaulayness of these two rings has been studied extensively. Let (R, m) be a local ring and I an ideal of R. An ideal J contained in I is called a reduction of I if J $I^{n}$ = $I^{n+1}$ for some integer n.geq.0. A reduction J of I is called a minimal reduction of I. The reduction number of I with respect to J is defined by (Fig.) S. Goto and Y.Shimoda characterized the Cohen-Macaulay property of the Rees algebra of the maximal ideal of a Cohen-Macaulay local ring in terms of the Cohen-Macaulay property of the associated graded ring of the maximal ideal and the reduction number of that maximal ideal. Let us state their theorem.m.m.

• Bulletin of the Korean Mathematical Society
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• v.57 no.2
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• pp.407-417
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• 2020

Let (R, m) be a d-dimensional Cohen-Macaulay local ring with infinite residue field. Let I be an ideal of R that has analytic spread ℓ(I) = d, satisfies the G_d condition, the weak Artin-Nagata property AN^-_d-2 and m is not an associated prime of R/I. In this paper, we show that if j₁(I) = λ(I/J) + λ[R/(J_d-1 :_RI+(J_d-2 :_RI+I):_R m^∞)] + 1, then I has almost minimal j-multiplicity, G(I) is Cohen-Macaulay and r_J(I) is at most 2, where J = (x₁, _…, x_d) is a general minimal reduction of I and J_i = (x₁, _…, x_i). In addition, the last theorem is in the spirit of a result of Sally who has studied the depth of associated graded rings and minimal reductions for m-primary ideals.

• Korean Journal of Mathematics
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• v.1
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• pp.27-32
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• 1993

• Journal of the Korean Mathematical Society
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• v.43 no.4
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• pp.871-883
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• 2006

We show that the Auslander index is the same as the column invariant over Gorenstein local rings. We also show that Ding's conjecture ([13]) holds for an isolated non-Gorenstein ring A satisfying a certain condition which seems to be weaker than the condition that the associated graded ring of A is Cohen-Macaulay.

• Bulletin of the Korean Mathematical Society
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• v.34 no.1
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• pp.35-42
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• 1997

Throughout this paper, all rings are assumed to be commutative with identity. By a local ring (R,m), we mean a Noetherian ring R which has a unique maximal ideal m. Let I be an ideal in a ring R and t an indeterminate over R.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.4
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• pp.799-809
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• 2013

An ideal I of a local ring (R, m, k) is said to be m-full if there exists an element $x{\in}m$ such that Im : x = I. An ideal I of a local ring R is said to have the Rees property if ${\mu}$(I) > ${\mu}$(J) for any ideal J containing I. We study properties of m-full ideals and we characterize m-primary m-full ideals in terms of the minimal number of generators of the ideals. In particular, for a m-primary ideal I of a 2-dimensional regular local ring (R, m, k), we will show that the following conditions are equivalent. 1. I is m-full 2. I has the Rees property 3. ${\mu}$(I)=o(I)+1 In this paper, let (R, m, k) be a commutative Noetherian local ring with infinite residue field k = R/m.

• Bulletin of the Korean Mathematical Society
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• v.39 no.3
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• pp.479-484
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• 2002

Let I be an ideal in a Gorenstein local ring A with the maximal ideal m and d = dim A. Then we say that I is a good ideal in A, if I contains a reduction $Q=(a_1,a_2,...,a_d)$ generated by d elements in A and $G(I)=\bigoplus_{n\geq0}I^n/I^{n+1}$ of I is a Gorenstein ring with a(G(I)) = 1-d, where a(G(I)) denotes the a-invariant of G(I). Let S = A[Q/a$_1$] and P = mS. In this paper, we show that the following conditions are equivalent. (1) $I^2$ = QI and I = Q:I. (2) $I^2S$ = $a_1$IS and IS = $a_1$S：sIS. (3) $I^2$Sp = $a_1$ISp and ISp = $a_1$Sp ：sp ISp. We denote by $X_A(Q)$ the set of good ideals I in $X_A(Q)$ such that I contains Q as a reduction. As a Corollary of this result, we show that $I\inX_A(Q)\Leftrightarrow\IS_P\inX_{SP}(Qp)$.