• Title, Summary, Keyword: local ring

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ON WEAKLY LOCAL RINGS

  • Piao, Zhelin;Ryu, Sung Ju;Sung, Hyo Jin;Yun, Sang Jo
    • Korean Journal of Mathematics
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    • v.28 no.1
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    • pp.65-73
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    • 2020
  • This article concerns a property of local rings and domains. A ring R is called weakly local if for every a ∈ R, a is regular or 1-a is regular, where a regular element means a non-zero-divisor. We study the structure of weakly local rings in relation to several kinds of factor rings and ring extensions that play roles in ring theory. We prove that the characteristic of a weakly local ring is either zero or a power of a prime number. It is also shown that the weakly local property can go up to polynomial (power series) rings and a kind of Abelian matrix rings.

STRONGLY CLEAN MATRIX RINGS OVER NONCOMMUTATIVE LOCAL RINGS

  • Li, Bingjun
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.1
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    • pp.71-78
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    • 2009
  • An element of a ring R with identity is called strongly clean if it is the sum of an idempotent and a unit that commute, and R is called strongly clean if every element of R is strongly clean. Let R be a noncommutative local ring, a criterion in terms of solvability of a simple quadratic equation in R is obtained for $M_2$(R) to be strongly clean.

THE SET OF ATTACHED PRIME IDEALS OF LOCAL COHOMOLOGY

  • RASOULYAR, S.
    • Honam Mathematical Journal
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    • v.23 no.1
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    • pp.1-4
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    • 2001
  • In [2, 7.3.2], the set of attached prime ideals of local cohomology module $H_m^n(M)$ were calculated, where (A, m) be Noetherian local ring, M finite A-module and $dim_A(M)=n$, and also in the special case in which furthermore A is a homomorphic image of a Gornestien local ring (A', m') (see [2, 11.3.6]). In this paper, we shall obtain this set, by another way in this special case.

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SOME REMARKS ON TYPES OF NOETHERIAN LOCAL RINGS

  • Lee, Kisuk
    • Journal of the Chungcheong Mathematical Society
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    • v.27 no.4
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    • pp.625-633
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    • 2014
  • We study some results which concern the types of Noetherian local rings, and improve slightly the previous result: For a complete unmixed (or quasi-unmixed) Noetherian local ring A, we prove that if either $A_p$ is Cohen-Macaulay, or $r(Ap){\leq}depth$ $A_p+1$ for every prime ideal p in A, then A is Cohen-Macaulay. Also, some analogous results for modules are considered.

A NOTE ON TYPES OF NOETHERIAN LOCAL RINGS

  • Lee, Kisuk
    • Bulletin of the Korean Mathematical Society
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    • v.39 no.4
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    • pp.645-652
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    • 2002
  • In this note we investigate some results which concern the types of local rings. In particular it is shown that if the type of a quasi-unmixed local ring A is less than or equal to depth A + 1, and $\hat{A}_p$ is Cohen-Macaulay for every prime $p\neq\hat{m}$, then A is Cohen-Macaulay. (This implies the previously known result: if A satisfies $(S_{n-1})}$, where n is the type of a .ins A, then A is Cohen-Macaulay.)

ON TYPES OF NOETHERIAN LOCAL RINGS AND MODULES

  • Lee, Ki-Suk
    • Journal of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.987-995
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    • 2007
  • We investigate some results which concern the types of Noetherian local rings. In particular, we show that if r(Ap) ${\le}$ depth Ap + 1 for each prime ideal p of a quasi-unmixed Noetherian local ring A, then A is Cohen-Macaulay. It is also shown that the Kawasaki conjecture holds when dim A ${\le}$ depth A + 1. At the end, we deal with some analogous results for modules, which are derived from the results studied on rings.

QUASIPOLAR MATRIX RINGS OVER LOCAL RINGS

  • Cui, Jian;Yin, Xiaobin
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.3
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    • pp.813-822
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    • 2014
  • A ring R is called quasipolar if for every a 2 R there exists $p^2=p{\in}R$ such that $p{\in}comm^2{_R}(a)$, $ a+p{\in}U(R)$ and $ap{\in}R^{qnil}$. The class of quasipolar rings lies properly between the class of strongly ${\pi}$-regular rings and the class of strongly clean rings. In this paper, we determine when a $2{\times}2$ matrix over a local ring is quasipolar. Necessary and sufficient conditions for a $2{\times}2$ matrix ring to be quasipolar are obtained.

A Study on the Control of the Beat Clarity and the Beat Period in a Ring Structure (링 구조물의 맥놀이의 선명도와 맥놀이 주기 조절에 관한 연구)

  • Kim, S.H.;Cui, C. X.
    • Transactions of the Korean Society for Noise and Vibration Engineering
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    • v.18 no.11
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    • pp.1170-1176
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    • 2008
  • In this study, we propose a new method to control both the beat clarity and beat period in a ring structure. An equivalent ring which satisfies the measured mode condition is determined by using the equivalent ring theory. Theoretical analysis and finite element analysis on the equivalent ring are performed to investigate the effect of the local structural modification on the beat clarity and beat period. Beat clarity and period are improved by attaching asymmetric mass or decreasing local thickness. Through the analysis on the equivalent ring, the proper position and the amount of the local variation are determined to satisfy the required clarity and period condition. All the analysis results are compared and verified by the experiment.

CHOW GROUPS OF COMPLETE REGULAR LOCAL RINGS III

  • Lee, Si-Chang
    • Communications of the Korean Mathematical Society
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    • v.17 no.2
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    • pp.221-227
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    • 2002
  • In this paper we will show that the followings ; (1) Let R be a regular local ring of dimension n. Then $A_{n-2}$(R) = 0. (2) Let R be a regular local ring of dimension n and I be an ideal in R of height 3 such that R/I is a Gorenstein ring. Then [I] = 0 in $A_{n-3}$(R). (3) Let R = V[[ $X_1$, $X_2$, …, $X_{5}$ ]]/(p+ $X_1$$^{t1}$ + $X_2$$^{t2}$ + $X_3$$^{t3}$ + $X_4$$^2$+ $X_{5}$ $^2$/), where p $\neq$2, $t_1$, $t_2$, $t_3$ are arbitrary positive integers and V is a complete discrete valuation ring with (p) = mv. Assume that R/m is algebraically closed. Then all the Chow group for R is 0 except the last Chow group.group.oup.

GENTRAL SEPARABLE ALGEBRAS OVER LOCAL-GLOBAL RINGS I

  • Kim, Jae-Gyeom
    • Bulletin of the Korean Mathematical Society
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    • v.30 no.1
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    • pp.61-64
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    • 1993
  • In this paper, we show that if R is a local-global domain then the Question holds. McDonald and Waterhouse in [6] and Estes and Guralnick in [5] introduced the concept of local-global rings (so called rings with many units) independently. A local-global ring is a commutative ring R with 1 satisfying; if a polynomial f in R[ $x_{1}$, .., $x_{n}$] represents a unit over $R_{P}$ for every maximal ideal P in R, then f represents a unit over R. Such rings include semilocal rings, or more generally, rings which are von Neumann regular mod their Jacobson radical, and the ring of all algebraic integers.s.s.

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