• Journal of the Chungcheong Mathematical Society
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• v.34 no.4
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• pp.335-343
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• 2021

In this article, we investigate the finiteness of Auslander Index when a ring A has not necessarily a canonical module, or a Gorenstein module. We also study the relations between column invariants and row invariants.

• Bulletin of the Korean Mathematical Society
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• v.57 no.3
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• pp.803-813
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• 2020

Let M be a finitely generated G-projective R-module over a PVMD R. We prove that M is projective if and only if the canonical map θ : M⨂_R M^∗ → Hom_R(Hom_R(M, M), R) is a surjective homomorphism. Particularly, if G-gldim(R) ⩽ ∞ and Extⁱ_R(M, M) = 0 (i ⩾ 1), then M is projective.

• Korean Journal of Mathematics
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• v.7 no.1
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• pp.71-77
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• 1999

We show that if the stable rank of $B^{\alpha}$ is one, then the stable rank of B is less than or equal to the order of G for any action of a finite group G. Also we give a short proof to the known fact that if the action of a finite group on a $C^*$-algebra B is saturated then the canonical conditional expectation from B to $B^{\alpha}$ is of index-finite type and the crossed product $C^*$-algebra is isomorphic to the algebra of compact operators on the Hilbert $B^{\alpha}$-module B.

• Honam Mathematical Journal
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• v.28 no.4
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• pp.533-541
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• 2006

let E be a Finsler modules over $C^*$-algebras. A with norm-map $\rho$ and L(E) set of all A-linear bonded operators on E. We show that the canonical homomorphism ${\phi}:L(E){\rightarrow}L(E_I)$ sending each operator T to its restriction $T|E_I$ is injective if and only if I is an essential ideal in the underlying $C^*$-algebra A. We also show that $T{\in}L(E)$ is a bounded below if and only if ${\mid}{\mid}x{\mid}{\mid}={\mid}{\mid}{\rho}{\prime}(x){\mid}{\mid}$ is complete, where ${\rho}{\prime}(x)={\rho}(Tx)$ for all $x{\in}E$. Also, we give a necessary and sufficient condition for the equivalence of the norms generated by the norm map.

• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.989-996
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• 2012

Let R be a commutative Noetherian ring and let I be an ideal of R. In this paper we study the amalgamated duplication ring $R{\bowtie}I$ which is introduced by D'Anna and Fontana. It is shown that if R is generically Cohen-Macaulay (resp. generically Gorenstein) and I is generically maximal Cohen-Macaulay (resp. generically canonical module), then $R{\bowtie}I$ is generically Cohen-Macaulay (resp. generically Gorenstein). We also de ned generically quasi-Gorenstein ring and we investigate when $R{\bowtie}I$ is generically quasi-Gorenstein. In addition, it is shown that $R{\bowtie}I$ is approximately Cohen-Macaulay if and only if R is approximately Cohen-Macaulay, provided some special conditions. Finally it is shown that if R is approximately Gorenstein, then $R{\bowtie}I$ is approximately Gorenstein.

• Bulletin of the Korean Mathematical Society
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• v.57 no.2
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• pp.275-280
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• 2020

Let (R, m) be a Noetherian local Cohen-Macaulay ring and I be a proper ideal of R. Assume that β_R(I, R) denotes the constant value of depth_R(R/Iⁿ) for n ≫ 0. In this paper we introduce the new notion γ_R(I, R) and then we prove the following inequalities: β_R(I, R) ≤ γ_R(I, R) ≤ dim R - cd(I, R) ≤ dim R/I. Also, some applications of these inequalities will be included.

• Proceedings of the Korean Society Of Semiconductor Equipment Technology
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• 2007.06a
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• pp.230-233
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• 2007

A system of anodic process of aluminum thin film has implemented for nanofabrication. The manufactured equipment consists of three main parts: chiller, reaction bath and power supply. The chiller module consists of refrigeration compressor, copper tube and coolant with a thermostat. The reaction bath has kept in same temperature as a thermodynamic canonical ensemble system during the anodic reaction process. The magnetic bar has stirred oxalic acid in bath for uniform reaction. The DC power supply has applied into two electrodes, aluminum for anode and platinum for cathode in the oxalic acid. The anodization process results in the formation of nanoporous thin films.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.21 no.3
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• pp.568-577
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• 2017

IEEE Std 1609.2 was defined for the support of communication security functions in the WAVE (Wireless Access in Vehicular Environments) system. IEEE Std 1609.2 defined the message structures of the security services and managements on the vehicular communication by using ASN.1 (Abstract Syntax Notation One). Also, this security message structures shall be encoded using the COER (Canonical Octet Encoding Rules). In this paper, we designed and implemented the IEEE Std 1609.2 message encoder/decoder handling the security messages defined in IEEE Std 1609.2. The designed encoder/decoder consists of three modules as follows : a module generating the message of C language data structures in accord with IEEE Std 1609.2 message structures, a message encoder module, a message decoder module. And the encoder/decoder was implemented on the Linux environment. Also we analyzed the performance by measuring the performance speed of the encoder/decoder implemented.

• 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}$._{A}$./.

• Molecules and Cells
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• v.41 no.9
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• pp.842-852
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• 2018

Notch signaling is an evolutionarily conserved pathway and involves in the regulation of various cellular and developmental processes. Ligand binding releases the intracellular domain of Notch receptor (NICD), which interacts with DNA-bound CSL [CBF1/Su(H)/Lag-1] to activate transcription of target genes. In the absence of NICD binding, CSL down-regulates target gene expression through the recruitment of various corepressor proteins including SMRT/NCoR (silencing mediator of retinoid and thyroid receptors/nuclear receptor corepressor), SHARP (SMRT/HDAC1-associated repressor protein), and KyoT2. Structural and functional studies revealed the molecular basis of these interactions, in which NICD coactivator and corepressor proteins competitively bind to ${\beta}-trefoil$ domain (BTD) of CSL using a conserved ${\varphi}W{\varphi}P$ motif (${\varphi}$ denotes any hydrophobic residues). To date, there are conflicting ideas regarding the molecular mechanism of SMRT-mediated repression of CSL as to whether CSL-SMRT interaction is direct or indirect (via the bridge factor SHARP). To solve this issue, we mapped the CSL-binding region of SMRT and employed a 'one- plus two-hybrid system' to obtain CSL interaction-defective mutants for this region. We identified the CSL-interaction module of SMRT (CIMS; amino acid 1816-1846) as the molecular determinant of its direct interaction with CSL. Notably, CIMS contains a canonical ${\varphi}W{\varphi}P$ sequence (APIWRP, amino acids 1832-1837) and directly interacts with CSL-BTD in a mode similar to other BTD-binding corepressors. Finally, we showed that CSL-interaction motif, rather than SHARP-interaction motif, of SMRT is involved in transcriptional repression of NICD in a cell-based assay. These results strongly suggest that SMRT participates in CSL-mediated repression via direct binding to CSL.