• Bulletin of the Korean Mathematical Society
• /
• v.59 no.5
• /
• pp.1237-1246
• /
• 2022

In this paper, we introduce the concept of N-pure ideal as a generalization of pure ideal. Using this concept, a new and interesting type of rings is presented, we call it a mid ring. Also, we provide new characterizations for von Neumann regular and zero-dimensional rings. Moreover, some results about mp-ring are given. Finally, a characterization for mid rings is provided. Then it is shown that the class of mid rings is strictly between the class of reduced mp-rings (p.f. rings) and the class of mp-rings.

• Bulletin of the Korean Mathematical Society
• /
• v.61 no.3
• /
• pp.763-777
• /
• 2024

Let R be a commutative ring with identity 1≠ 0. In this paper, we continue the study started in [10] to further investigate when the extended zero-divisor graph of R, denoted as $\bar{\Gamma}$(R), is complemented. We also study when $\bar{\Gamma}$(R) is uniquely complemented. We give a complete characterization of when $\bar{\Gamma}$(R) of a finite ring R is complemented. Various examples are given using the direct product of rings and idealizations of modules.

• Kyungpook Mathematical Journal
• /
• v.61 no.4
• /
• pp.823-830
• /
• 2021

Let {R_i}_i∈I be an infinite family of rings and R = ∏_i∈I R_i their product. In this paper, we investigate the prime spectrum of R by 𝓕-limits. Special attention is paid to relationship between the elements of Spec(R_i) and the elements of Spec(∏_i∈I R_i) use 𝓕-lim, also we give a new condition so that ∏_i∈I R_i is a zero dimensional ring.

• Bulletin of the Korean Mathematical Society
• /
• v.20 no.1
• /
• pp.45-49
• /
• 1983

If R is ring and M is a right (or left) R-module, then M is called a faithful R-module if, for some a in R, x.a=0 for all x.mem.M then a=0. In [4], R.E. Johnson defines that M is a prime module if every non-zero submodule of M is faithful. Let us define that M is of prime type provided that M is faithful if and only if every non-zero submodule is faithful. We call a right (left) ideal I of R is of prime type if R/I is of prime type as a R-module. This is equivalent to the condition that if xRy.subeq.I then either x.mem.I ro y.mem.I (see [5:3:1]). It is easy to see that in case R is a commutative ring then a right or left ideal of a prime type is just a prime ideal. We have defined in [5], that a chain of right ideals of prime type in a ring R is a finite strictly increasing sequence I$_{0}$.contnd.I$_{1}$.contnd....contnd.I$_{n}$; the length of the chain is n. By the right dimension of a ring R, which is denoted by dim, R, we mean the supremum of the length of all chains of right ideals of prime type in R. It is an integer .geq.0 or .inf.. The left dimension of R, which is denoted by dim$_{l}$ R is similarly defined. It was shown in [5], that dim$_{r}$R=0 if and only if dim$_{l}$ R=0 if and only if R modulo the prime radical is a strongly regular ring. By "a strongly regular ring", we mean that for every a in R there is x in R such that axa=a=a$^{2}$x. It was also shown that R is a simple ring if and only if every right ideal is of prime type if and only if every left ideal is of prime type. In case, R is a (right or left) primitive ring then dim$_{r}$R=n if and only if dim$_{l}$ R=n if and only if R.iden.D$_{n+1}$ , n+1 by n+1 matrix ring on a division ring D. in this paper, we establish the following results: (1) If R is prime ring and dim$_{r}$R=n then either R is a righe Ore domain such that every non-zero right ideal of a prime type contains a non-zero minimal prime ideal or the classical ring of ritght quotients is isomorphic to m*m matrix ring over a division ring where m.leq.n+1. (b) If R is prime ring and dim$_{r}$R=n then dim$_{l}$ R=n if dim$_{l}$ R=n if dim$_{l}$ R<.inf. (c) Let R be a principal right and left ideal domain. If dim$_{r}$R=1 then R is an unique factorization domain.TEX>R=1 then R is an unique factorization domain.

• Korean Journal of Optics and Photonics
• /
• v.7 no.1
• /
• pp.66-71
• /
• 1996

We analyzed and fabricated InP/InGaAs avalanche photodiode (APD) having floating guard ring (FGR). Since the FGR-APD is very simple to fabricate and highly reliable, the fabrication of FGR-APD and its application to the optical receiver are very useful and interesting. A double zinc diffusion was employed to fabricate and one dimensional electric field analysis was used for design. Two dimensional gain measurement showed that the FGR suppressed gain at the curved edge, indicating the successful behavior as a guard ring. The fabricated device had 35 GHz of gain-bandwidth product, and showed the sensitivity of -31.9 dBm at a bit error rate of $10^{-9}$ when it was applied to a 2.5 Gbps optical receiver.

• Bulletin of the Korean Mathematical Society
• /
• v.51 no.1
• /
• pp.253-266
• /
• 2014

Commutative rings in which every prime ideal is an intersection of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that classical prime submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are classical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings R over which all modules are classical Hilbert are characterized. Furthermore, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.

• Communications of the Korean Mathematical Society
• /
• v.39 no.3
• /
• pp.563-574
• /
• 2024

Given a sequence of point blow-ups of smooth n-dimensional projective varieties Z_i defined over an algebraically closed field $k,\,Z_s\,{\overset{{\pi}_s}{\longrightarrow}}\,Z_{s-1}\,{\overset{{\pi}_{s-1}}{\longrightarrow}}\,{\cdots}\,{\overset{{\pi}_2}{\longrightarrow}}\,Z_1\,{\overset{{\pi}_1}{\longrightarrow}}\,Z_0$, with Z₀ ≅ ℙⁿ, we give two presentations of the Chow ring A^•(Z_s) of its sky. The first one uses the classes of the total transforms of the exceptional components as generators and the second one uses the classes of the strict transforms ones. We prove that the skies of two sequences of point blow-ups of the same length have isomorphic Chow rings. Finally we give a characterization of the final divisors of a sequence of point blow-ups in terms of some relations defined over the Chow group of zero-cycles A₀(Z_s) of its sky.

• Journal of the Chungcheong Mathematical Society
• /
• v.22 no.4
• /
• pp.777-787
• /
• 2009

In this paper we investigate the upper bound on the Castelnuovo-Mumford regularity of a graded module with linear free presentation. Let M be a finitely generated graded module over a polynomial ring R with zero dimensional support. We prove that if M is generated by elements of degree $d{\geq}0$ with a linear free presentation $$\bigoplus^p{R}(-d-1)\longrightarrow^{\phi}\bigoplus^q{R}(-d){\longrightarrow}M{\longrightarrow}0$$, then the Castelnuovo-Mumford regularity of M is at most d+q-1. As an important application, we can prove vector bundle technique, which was used in [11], [13], [17] as a tool for obtaining several remarkable results.

• Journal of the Korean Mathematical Society
• /
• v.54 no.5
• /
• pp.1379-1410
• /
• 2017

Kang and Ko introduced a skew-symmetrizable matrix to describe a structure theorem for complete intersections of grade 4. Let $R=k[w_0,\;w_1,\;w_2,\;{\ldots},\;w_m]$ be the polynomial ring over an algebraically closed field k with indetermiantes $w_l$ and deg $w_l=1$, and $I_i$ a homogeneous perfect ideal of grade 3 with type $t_i$ defined by a skew-symmetrizable matrix $G_i(1{\leq}t_i{\leq}4)$. We show that for m = 2 the Hilbert function of the zero dimensional standard k-algebra $R/I_i$ is determined by CI-sequences and a Gorenstein sequence. As an application of this result we show that for i = 1, 2, 3 and for m = 3 a Gorenstein sequence $h(R/H_i)=(1,\;4,\;h_2,\;{\ldots},\;h_s)$ is unimodal, where $H_i$ is the sum of homogeneous perfect ideals $I_i$ and $J_i$ which are geometrically linked by a homogeneous regular sequence z in $I_i{\cap}J_i$.

• Journal of Mechanical Science and Technology
• /
• v.17 no.12
• /
• pp.1835-1843
• /
• 2003

This paper introduces the development of the caliper system for a geometry PIG (Pipeline Inspection Gauge). The objective of the caliper system is to detect and measure dents, wrinkles, and ovalities affect the pipe structural integrity. The developed caliper system consists of a finger arm, an anisotropic permanent magnet, a back yoke, pins, pinholes and a linear hall effect sensor. The angle displacement of the finger arm is measured by the change of the magnetic field in sensing module. Therefore the sensitivity of the caliper system mainly depends on the magnitude of the magnetic field inside the sensing module. In this research, the ring shaped anisotropic permanent magnet and linear hall effect sensors were used to produce and measure the magnetic field. The structure of the permanent magnet, the back yoke and pinhole positions were optimized that the magnitude of the magnetic field range between a high of 0.1020 Tesla and a low of zero by using three dimensional nonlinear finite element methods. A simulator was fabricated to prove the effectiveness of the developed caliper system and the computational scheme using the finite element method. The experimental results show that the developed caliper system is quite efficient for the geometry PIG with good performance.