• Journal for History of Mathematics
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• v.19 no.2
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• pp.125-140
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• 2006

In 1933, I. Schur introduced a Schur ring in connection with permutation group and regular subgroup. After then, it was studied mostly for purely group theoretical purposes. In 1970s, Klin and Poschel initiated its usage in the investigation of graphs, especially for Cayley and circulant graphs. Nowadays it is known that Schur ring is one of the best way to enumerate Cayley graphs. In this paper we study the origin of Schur ring back to 1933 and keep trace its evolution to graph theory and combinatorics.

• Honam Mathematical Journal
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• v.23 no.1
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• pp.5-14
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• 2001

Let E be an injective module over a commutative Noetherian ring A. In this paper we will show that if I is regular ideal, then the sequence of sets $$Ass_A((I^n)^{{\star}(E)}/I^n),\;n{\in}N$$ is ultimately constant. Also we obtain some related results. (Here for an ideal J of A, $J^{{\star}(E)}$ denotes the integral closure of J relative to E.

• East Asian mathematical journal
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• v.28 no.3
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• pp.281-285
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• 2012

In this paper, the concept of *-prime ideals in non-associative near-rings is introduced and then will be studied. For this purpose, first we introduce the notions of *-operation, *-prime ideal and *-system in a near-ring. Next, we will define the *-sequence, *-strongly nilpotent *-prime radical of near-rings, and then obtain some characterizations of *-prime ideal and *-prime radical $r_s$(I) of an ideal I of near-ring N.

• Kyungpook Mathematical Journal
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• v.58 no.3
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• pp.481-488
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• 2018

Let I be a ${\ast}$-ideal on a 2-torsion free ${\ast}$-prime ring and $F:R{\rightarrow}R$ a generalized derivation with an associated derivation $d:R{\rightarrow}R$. The aim of this paper is to explore the condition under which generalized derivation F becomes a left centralizer i.e., associated derivation d becomes a trivial map (i.e., zero map) on R.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.147-160
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• 2024

In this paper, we define two new classes of monomial ideals I_𝑙,d and J_k,d. When d ≥ 2k + 1 and 𝑙 ≤ d - k - 1, we give the exact formulas to compute the depth and Stanley depth of quotient rings S/I^t_𝑙,d for all t ≥ 1. When d = 2k = 2𝑙, we compute the depth and Stanley depth of quotient ring S/I_𝑙,d. When d ≥ 2k, we also compute the depth and Stanley depth of quotient ring S/J_k,d.

• Bulletin of the Korean Mathematical Society
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• v.31 no.2
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• pp.163-165
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• 1994

Modules which satisfy the converse of Schur's lemma have been studied by many authors. In [6], R. Ware proved that a projective module P over a semiprime ring R is irreducible if and only if En $d_{R}$(P) is a division ring. Also, Y. Hirano and J.K. Park proved that a torsionless module M over a semiprime ring R is irreducible if and only if En $d_{R}$(M) is a division ring. In case R is a commutative ring, we obtain the following: An R-module M is irreducible if and only if En $d_{R}$(M) is a division ring and M is a multiplication R-module. Throughout this paper, R is commutative ring with identity and all modules are unital left R-modules. Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for each submodule N of M, there exists and ideal I of R such that N=IM. Cyclic R-modules are multiplication modules. In particular, irreducible R-modules are multiplication modules.dules.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.595-613
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• 2022

Let R be a commutative ring with identity and let I be a proper ideal of R. In this paper, we study the ideal based extended zero-divisor graph 𝚪'_I (R) and prove that 𝚪'_I (R) is connected with diameter at most two and if 𝚪'_I (R) contains a cycle, then girth is at most four girth at most four. Furthermore, we study affinity the connection between the ideal based extended zero-divisor graph 𝚪'_I (R) and the ideal-based zero-divisor graph 𝚪_I (R) associated with the ideal I of R. Among the other things, for a radical ideal of a ring R, we show that the ideal-based extended zero-divisor graph 𝚪'_I (R) is identical to the ideal-based zero-divisor graph 𝚪_I (R) if and only if R has exactly two minimal prime-ideals which contain I.

• The Transactions of the Korean Institute of Electrical Engineers D
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• v.51 no.12
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• pp.562-570
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• 2002

The Purpose of this paper is to design and manufacture an educational system in order to demonstrate the causes and effects of electromagnetic induction.'rho educational system described in this study is a "jumping ring apparatus". This system demonstrates the principle of electromagnetic induction, a force from AC sources, Lenz's law of repulsion and transformer. The educational system is composed of a jumping ring apparatus, a sensor array, encoder, A/D converter, D/A converter and nonlinear controller. The educational system is controlled by 586 PC using Turbo C program. The sensor array is composed of 20 optical sensors. The nonlinear controller consists of nonlinear control algorithm and control board included SCR, FET and phase controller. The A/D converter is used to show the height of ring position to analog for an education purpose. The control signal calculated from the nonlinear control of algorithm send control board through 8 bit D/A convertor. Experiment results are given to verify that Proposed nonlinear controller is useful in on line control of the educational system.al system.

• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.867-871
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• 2009

A module M over a ring R is said to satisfy (P) if every generating set of M contains an independent generating set. The following results are proved; (1) Let $\tau$ = ($\mathbb{T}_\tau,\;\mathbb{F}_\tau$) be a hereditary torsion theory such that $\mathbb{T}_\tau$ $\neq$ Mod-R. Then every $\tau$-torsionfree R-module satisfies (P) if and only if S = R/$\tau$(R) is a division ring. (2) Let $\mathcal{K}$ be a hereditary pre-torsion class of modules. Then every module in $\mathcal{K}$ satisfies (P) if and only if either $\mathcal{K}$ = {0} or S = R/$Soc_\mathcal{K}$(R) is a division ring, where $Soc_\mathcal{K}$(R) = $\cap${I 4\leq$ $R_R$ : R/I$\in\mathcal{K}$}.

• Journal of the Korean Institute of Telematics and Electronics
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• v.24 no.6
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• pp.1040-1048
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• 1987

We propose a hardware architecture called Multiring which is applicable for various geometrical operations on rectilinear objects such as design rule checking in VLSI layout and many image processing operations including noise suppression and coutour extraction. It has both a fast execution speed and extremely high flexibility. The whole architecture is mainly divided into four parts` I/O between host and Multiring, ring memory, linear processor array and instruction decoder. Data transmission between host and Multiring is bit serial thereby reducing the bandwidth requirement for teh channel and the number of external pins, while each row data in the bit map stored in ring memory is processed in the corresponding processor in full parallelism. Each processor is simultaneously configured by the instruction decoder/controller to perform one of the 16 basic instructions such as Boolean (AND, OR, NOT, and Copy), geometrical(Expand and Shrink), and I/O operations each ring cycle, which gives Multiring maximal flexibility in terms of design rule change or the instruction set enhancement. Correct functional behavior of Multiring was confirmed by successfully running a software simulator having one-to-one structural correspondence to the Multiring hardware.