• Kyungpook Mathematical Journal
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• v.63 no.3
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• pp.333-344
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• 2023

In this paper, we generalize the notions uniformly S-flat, briefly u-S-flat, modules and dimensions. We introduce and study the notions of weak u-S-flat modules. An R-module M is said to be weak u-S-flat if Tor^R₁ (R/I, M) is u-S-torsion for any ideal I of R. This new class of modules will be used to characterize u-S-von Neumann regular rings. Hence, we introduce the weak u-S-flat dimensions of modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed.

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.711-720
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• 2021

Let R denote a commutative noetherian ring, and let 𝐱 := x1, …, xd be an R-regular sequence. Suppose that 𝖆 denotes a monomial ideal with respect to 𝐱. The first purpose of this article is to show that 𝖆 is irreducible if and only if 𝖆 is a generalized-parametric ideal. Next, it is shown that, for any integer n ≥ 1, (x1, …, xd)n = ⋂P(f), where the intersection (irredundant) is taken over all monomials f = xe11 ⋯ xedd such that deg(f) = n - 1 and P(f) := (xe1+11, ⋯, xed+1d). The second main result of this paper shows that if 𝖖 := (𝐱) is a prime ideal of R which is contained in the Jacobson radical of R and R is 𝖖-adically complete, then 𝖆 is a parameter ideal if and only if 𝖆 is a monomial irreducible ideal and Rad(𝖆) = 𝖖. In addition, if a is generated by monomials m1, …, mr, then Rad(𝖆), the radical of a, is also monomial and Rad(𝖆) = (ω1, …, ωr), where ωi = rad(mi) for all i = 1, …, r.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.679-687
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• 2005

In this paper we shall give a new definition for a quasi-power module P(M) and discuss some properties for P(M). The quasi-power module P(M) is a direct sum of invertible quasi-submodules C(H)'s of P(M) and then the quasi-submodule C(H) is also a direct sum of strongly cyclic quasi-submodules of C(H). When M is a quasi-perfect right R-module, we shall see that the quasi-power module P(M) is invertible.

• Bulletin of the Korean Mathematical Society
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• v.47 no.3
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• pp.653-661
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• 2010

Let R be a ring, and let $\Psi$(R) be the ideal generated by the set {x $\in$R | 1 + sxt $\in$ R is unit-regular for all s, t $\in$ R}. We show that $\Psi$(R) has "radical-like" property. It is proven that $\Psi$(R) has stable range one. Thus, diagonal reduction of matrices over such ideal is reduced.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1855-1861
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• 2013

Let R be a commutative Noetherian ring, I an ideal of R, M and N two R-modules. We characterize the least integer i such that $H^i_I(M,N)$ is not weakly Artinian by using the notion of weakly filter regular sequences. Also, a local-global principle for minimax generalized local cohomology modules is shown and the result generalizes the corresponding result for local cohomology modules.

• Journal of the Korean Mathematical Society
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• v.47 no.1
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• pp.1-15
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• 2010

We introduce, in this article, the quasi-stable exchange ideal for associative rings. If I is a quasi-stable exchange ideal of a ring R, then so is $M_n$(I) as an ideal of $M_n$(R). As an application, we prove that every square regular matrix over quasi-stable exchange ideal admits a diagonal reduction by quasi invertible matrices. Examples of such ideals are given as well.

• The Pure and Applied Mathematics
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• v.25 no.2
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• pp.115-125
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• 2018

In this paper, we introduce the notion of a clean ordered Krasner hyperring and investigate some properties of it. Now, let (R, +, ${\cdot}$, ${\leq}$) be a clean ordered Krasner hyperring. The following is a natural question to ask: Is there a strongly regular relation ${\sigma}$ on R for which $R/{\sigma}$ is a clean ordered ring? Our motivation to write the present paper is reply to the above question.

• Kyungpook Mathematical Journal
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• v.63 no.1
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• pp.15-27
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• 2023

A purely extending module is a generalization of an extending module. In this paper, we study several properties of purely extending modules and introduce the notion of purely essentially Baer modules. A module M is said to be a purely essentially Baer if the right annihilator in M of any left ideal of the endomorphism ring of M is essential in a pure submodule of M. We study some properties of purely essentially Baer modules and characterize von Neumann regular rings in terms of purely essentially Baer modules.

• Journal of the Institute of Convergence Signal Processing
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• v.12 no.4
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• pp.341-347
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• 2011

This paper presents the performance evaluation. optimal design. and complex obstacle detection of an overlapped ultrasonic sensor ring by introducing a new concept of effective beam width. It is assumed that a set of ultrasonic sensors of the same type are arranged along a circle of nonzero radius at regular spacings with their beams overlapped. First, the global positional uncertainty of an overlapped ultrasonic sensor ring is expressed by the average value of local positional uncertainty over the entire obstacle detection range. The effective beam width of an overlapped ultrasonic sensor ring is assessed as the beam width of a single ultrasonic sensor having the same amount of global positional uncertainty, from which a normalized obstacle detection performance index is defined. Second. using the defined index, the design parameters of an overlapped ultrasonic sensor ring are optimized for minimal positional uncertainty in obstacle detection. For a given number of ultrasonic sensors, the optimal radius of an overlapped ultrasonic sensor ring is determined, and for a given radius of an overlapped ultrasonic sensor ring, the optimal number of ultrasonic sensors is determined. Third, the decision rules of positional uncertainty zone for multiple obstacle detection are provided based on the inequality relationships among obstacle distances by three adjacent ultrasonic sensors. Using the provided rules, the obstacle outline detection is performed in a rather complex environment consisting of several obstacles of different shapes.

• Journal of the Institute of Convergence Signal Processing
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• v.12 no.1
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• pp.67-75
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• 2011

This paper presents the systematic optimal design of an overlapped ultrasonic sensor ring for high performance obstacle detection using effective beam overlap. Basically, a set of low directivity ultrasonic sensors of the same type are arranged in a circle at regular intervals with their beams overlapped. First, both real and simplified beam patterns of an ultrasonic sensor and several sensor models for obstacle position estimation within its beam pattern are introduced. Second, the obstacle detection range of an overlapped ultrasonic sensor ring and its simple sensor model for obstacle position estimation are described. Third, for both conic and non-conic shaped beam pattern, the design indices of an overlapped ultrasonic sensor ring for minimal positional uncertainty in obstacle detection are defined. Fourth, the constraints imposed on the structural parameters of an overlapped ultrasonic sensor ring to guarantee non empty beam overlap and to avoid excessive beam overlap are derived. Fifth, the optimal number of ultrasonic sensors for a given radius of an overlapped ultrasonic sensor ring and the optimal radius of an overlapped ultrasonic sensor ring are determined. Throughout this paper, the MA40B8 from Murata Inc. is taken as a representative commercial low directivity ultrasonic sensor.