• Title/Summary/Keyword: A.I

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EQUIMULTIPLE GOOD IDEALS WITH HEIGHT 1

  • Kim, Mee-Kyoung
    • Journal of the Korean Mathematical Society
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    • v.39 no.1
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    • pp.127-135
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    • 2002
  • Let I be an ideal in a Gorenstein local ring A with the maximal ideal m. Then we say that I is an equimultiple good ideal in A, if I contains a reduction Q = ( $a_1$, $a_2$,ㆍㆍㆍ, $a_{s}$ ) generated by s elements in A and G(I) =(equation omitted)$_{n 0}$ $I^{n}$ / $I^{n+1}$ of I is a Gorenstein ring with a(G(I)) = 1 - s, where s = h $t_{A}$ I and a(G(I)) denotes the a-invariant of G(I). Let $X_{A}$$^{s}$ denote the set of equimultiple good ideals I in A with h $t_{A}$ I = s, R(I) = A [It] be the Rees algebra of I, and $K_{R(I)}$ denote the canonical module of R(I). Let a I such that $I^{n+l}$ = a $I^{n}$ for some n$\geq$0 and $\mu$$_{A}$(I)$\geq$2, where $\mu$$_{A}$(I) denotes the number of elements in a minimal system of generators of I. Assume that A/I is a Cohen-Macaulay ring. We show that the following conditions are equivalent. (1) $K_{R(I)}$(equation omitted)R(I)+as graded R(I)-modules. (2) $I^2$ = aI and aA : I$\in$ $X^1$$_{A}$._{A}$./.

ON DIRECT SUMS IN BOUNDED BCK-ALGEBRAS

  • HUANG YISHENG
    • Communications of the Korean Mathematical Society
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    • v.20 no.2
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    • pp.221-229
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    • 2005
  • In this paper we consider the decompositions of subdirect sums and direct sums in bounded BCK-algebras. The main results are as follows. Given a bounded BCK-algebra X, if X can be decomposed as the subdirect sum $\bar{\bigoplus}_{i{\in}I}A_i$ of a nonzero ideal family $\{A_i\;{\mid}\;i{\in}I\}$ of X, then I is finite, every $A_i$ is bounded, and X is embeddable in the direct sum $\bar{\bigoplus}_{i{\in}I}A_i$ ; if X is with condition (S), then it can be decomposed as the subdirect sum $\bar{\bigoplus}_{i{\in}I}A_i$ if and only if it can be decomposed as the direct sum $\bar{\bigoplus}_{i{\in}I}A_i$ ; if X can be decomposed as the direct sum $\bar{\bigoplus}_{i{\in}I}A_i$, then it is isomorphic to the direct product $\prod_{i{\in}I}A_i$.

AN ASSOCIATED SEQUENCE OF IDEALS OF AN INCREASING SEQUENCE OF RINGS

  • Ali, Benhissi;Abdelamir, Dabbabi
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.6
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    • pp.1349-1357
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    • 2022
  • Let 𝒜 = (An)n≥0 be an increasing sequence of rings. We say that 𝓘 = (In)n≥0 is an associated sequence of ideals of 𝒜 if I0 = A0 and for each n ≥ 1, In is an ideal of An contained in In+1. We define the polynomial ring and the power series ring as follows: $I[X]\, = \,\{\, f \,=\, {\sum}_{i=0}^{n}a_iX^i\,{\in}\,A[X]\,:\,n\,{\in}\,\mathbb{N},\,a_i\,{\in}\,I_i \,\}$ and $I[[X]]\, = \,\{\, f \,=\, {\sum}_{i=0}^{+{\infty}}a_iX^i\,{\in}\,A[[X]]\,:\,a_i\,{\in}\,I_i \,\}$. In this paper we study the Noetherian and the SFT properties of these rings and their consequences.

SELF-ADJOINT INTERPOLATION FOR OPERATORS IN TRIDIAGONAL ALGEBRAS

  • Kang, Joo-Ho;Jo, Young-Soo
    • Bulletin of the Korean Mathematical Society
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    • v.39 no.3
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    • pp.423-430
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    • 2002
  • Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for n-operators satisfies the equation $AX_{}i$ = $Y_{i}$ for i/ = 1,2,…, n. In this article, we obtained the following : Let X = ($x_{i\sigma(i)}$ and Y = ($y_{ij}$ be operators in B(H) such that $X_{i\sigma(i)}\neq\;0$ for all i. Then the following statements are equivalent. (1) There exists an operator A in Alg L such that AX = Y, every E in L reduces A and A is a self-adjoint operator. (2) sup ${\frac{\parallel{\sum^n}_{i=1}E_iYf_i\parallel}{\parallel{\sum^n}_{i=1}E_iXf_i\parallel}n\;\epsilon\;N,E_i\;\epsilon\;L and f_i\;\epsilon\;H}$ < $\infty$ and $x_{i,\sigma(i)}y_{i,\sigma(i)}$ is real for all i = 1,2, ....

SOME RESULTS ON 1-ABSORBING PRIMARY AND WEAKLY 1-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS

  • Nikandish, Reza;Nikmehr, Mohammad Javad;Yassine, Ali
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.5
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    • pp.1069-1078
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    • 2021
  • Let R be a commutative ring with identity. A proper ideal I of R is called 1-absorbing primary ([4]) if for all nonunit a, b, c ∈ R such that abc ∈ I, then either ab ∈ I or c ∈ ${\sqrt{1}}$. The concept of 1-absorbing primary ideals in a polynomial ring, in a PID and in idealization of a module is studied. Moreover, we introduce weakly 1-absorbing primary ideals which are generalization of weakly prime ideals and 1-absorbing primary ideals. A proper ideal I of R is called weakly 1-absorbing primary if for all nonunit a, b, c ∈ R such that 0 ≠ abc ∈ I, then either ab ∈ I or c ∈ ${\sqrt{1}}$. Some properties of weakly 1-absorbing primary ideals are investigated. For instance, weakly 1-absorbing primary ideals in decomposable rings are characterized. Among other things, it is proved that if I is a weakly 1-absorbing primary ideal of a ring R and 0 ≠ I1I2I3 ⊆ I for some ideals I1, I2, I3 of R such that I is free triple-zero with respect to I1I2I3, then I1I2 ⊆ I or I3 ⊆ I.

NOTE ON GOOD IDEALS IN GORENSTEIN LOCAL RINGS

  • Kim, Mee-Kyoung
    • 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 $I\inX_A(Q)\Leftrightarrow\IS_P\inX_{SP}(Qp)$.

SYSTEMS OF DERIVATIONS ON BANACH ALGEBRAS

  • Lee, Eun-Hwi
    • Communications of the Korean Mathematical Society
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    • v.12 no.2
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    • pp.251-256
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    • 1997
  • We show that a strong system of derivations ${D_0, D_1,\cdots,D_m}$ on a commutative Banach algebra A is contained in the radical of A if it satisfies one of the following conditions for separating spaces; (1) $\partial(D_i) \subseteq rad(A) and \partial(D_i) \subseteq K D_i(rad(A))$ for all i, where $K D_i(rad(A)) = {x \in rad(A))$ : for each $m \geq 1, D^m_i(x) \in rad(A)}$. (2) $(D^m_i) \subseteq rad(A)$ for all i and m. (3) $\bar{x\partial(D_i)} = \partial(D_i)$ for all i and all nonzero x in rad(A).

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EXPANSIVE HOMEOMORPHISMS WITH THE SHADOWING PROPERTY ON ZERO DIMENSIONAL SPACES

  • Park, Jong-Jin
    • Communications of the Korean Mathematical Society
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    • v.19 no.4
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    • pp.759-764
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    • 2004
  • Let X = {a} ${\cup}$ {$a_{i}$ ${$\mid$}$i $\in$ N} be a subspace of Euclidean space $E^2$ such that $lim_{{i}{\longrightarrow}{$\infty}}a_{i}$ = a and $a_{i}\;{\neq}\;a_{j}$ for $i{\neq}j$. Then it is well known that the space X has no expansive homeomorphisms with the shadowing property. In this paper we show that the set of all expansive homeomorphisms with the shadowing property on the space Y is dense in the space H(Y) of all homeomorphisms on Y, where Y = {a, b} ${\cup}$ {$a_{i}{$\mid$}i{\in}Z$} is a subspace of $E^2$ such that $lim_{i}$-$\infty$ $a_{i}$ = b and $lim_{{i}{\longrightarrow}{$\infty}}a_{i}$ = a with the following properties; $a_{i}{\neq}a_{j}$ for $i{\neq}j$ and $a{\neq}b$.

APPROXIMATION OF CUBIC MAPPINGS WITH n-VARIABLES IN β-NORMED LEFT BANACH MODULE ON BANACH ALGEBRAS

  • Gordji, Majid Eshaghi;Khodaei, Hamid;Najati, Abbas
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.5
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    • pp.1063-1078
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    • 2011
  • Let M = {1, 2, ${\ldots}$, n} and let V = {$I{\subseteq}M:1{\in}I$}. Denote M\I by $I^c$ for $I{\in}V$. The goal of this paper is to investigate the solution and the stability using the alternative fixed point of generalized cubic functional equation $ \sum\limits_{I{\in}V}f(\sum\limits_{i{\in}I}a_ix_i-\sum\limits_{i{\in}I^c}a_ix_i)=2{^{n-2}{a_1}}\sum\limits_{i=2}^na_i^2[f(x_1+x_i)+f(x_1-x_i)]+2{^{n-1}{a_1}(a^2_1-\sum\limits_{i=2}^2a^2_i)f(x_1)$ in ${\beta}$-Banach modules on Banach algebras, where $a_1,{\ldots},a_n{\in}\mathbb{Z}{\backslash}\{0\}$ with $a_1{\neq}={\pm}1$ and $a_n=1$.

Distribution von Pronomina in A.cI.-Konstruktionen (A.c.I.-구문에서의 대명사 분포에 관한 연구)

  • Kim Youn-Chan
    • Koreanishche Zeitschrift fur Deutsche Sprachwissenschaft
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    • v.7
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    • pp.105-125
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    • 2003
  • Personal- und Reflexivpronomen stehen in einer komplementaren Beziehung zueinander. Die vorliegende Arbeit zeigt, welches von beiden Pronomen verschiedenen AcI-Konstruktionen jeweils richtig ist Vor allem stelle ich hier die These auf, dass Grammatik in erster Linie ein Regelwerk ist und dass dernzufolge $S\"{a}tze$ mit gleicher Struktur immer einheitlich und konsistent mit denselben Regeln $erkl\"{a}rt$ werden sollten: (1) a. Der Gefangen $l\"{a}sst$ den Polizisten auf ${\ast}sich_i/ihn_i$ achten. b. $Hans_i{\;}l\"{a}sst$ Maria zu $sich_i/{\ast}ihm_i$ kommen. (2) a. $Peter_i$ sieht Hans von ${\ast}sich_i/ihm_i$ betrogen. b. $Hans_i{\;}f\"{u}hlt$ die Freundin von $sich_i/{\ast}ihm_i$ weggezogen. Bei diesen Beispielen liegt der Verteilungsunterschied der Pronomina nicht an Grammatikregeln sondern am individuellen Sprachgebrauch. Ferner wird hier darauf hingewiesen, dass die AcI-Verben mit Ausnahme des Verbs lassen a1s 3-wertige Verben so wie die $S\"{a}tze$ in Beispiel (3) zu behandein sind D.h., die Struktur $f\"{u}r$ AcI-Konstruktionen sieht so aus wie (4b), nicht wie (4a). Betrachten wir dies noch einmal an einem Beispiel: (3) a. Ich sehe mich $m\"{u}de$ b. Ich $f\"{u}hle$ mich viel besser.

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