• Korean Journal of Mathematics
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• 제29권3호
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• pp.483-492
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• 2021

This article concerns the property that for any element a in a ring, if a²ⁿ = aⁿ for some n ≥ 2 then a² = a. The class of rings with this property is large, but there also exist many kinds of rings without that, for example, rings of characteristic ≠2 and finite fields of characteristic ≥ 3. Rings with such a property is called reduced-over-idempotent. The study of reduced-over-idempotent rings is based on the fact that the characteristic is 2 and every nonzero non-identity element generates an infinite multiplicative semigroup without identity. It is proved that the reduced-over-idempotent property pass to polynomial rings, and we provide power series rings with a partial affirmative argument. It is also proved that every finitely generated subring of a locally finite reduced-over-idempotent ring is isomorphic to a finite direct product of copies of the prime field {0, 1}. A method to construct reduced-over-idempotent fields is also provided.

• 대한수학회보
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• 제59권6호
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• pp.1409-1422
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• 2022

A 3-fold quotient terminal singularity is of the type $\frac{1}{r}(b,1,-1)$ with gcd(r, b) = 1. In [6], it is proved that the economic resolution of a 3-fold terminal quotient singularity is isomorphic to a distinguished component of a moduli space 𝓜_𝜃 of 𝜃-stable G-constellations for a suitable 𝜃. This paper proves that each connected component of the moduli space 𝓜_𝜃 has a torus fixed point and classifies all torus fixed points on 𝓜_𝜃. By product, we show that for $\frac{1}{2k+1}(k+1,1,-1)$ case the moduli space 𝓜_𝜃 is irreducible.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제29권1호
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• pp.59-68
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• 2022

For a finite group G ⊂ GL(n, ℂ), the G-Hilbert scheme is a fine moduli space of G-clusters, which are 0-dimensional G-invariant subschemes Z with H⁰(𝒪_Z ) isomorphic to ℂ[G]. In many cases, the G-Hilbert scheme provides a good resolution of the quotient singularity ℂⁿ/G, but in general it can be very singular. In this note, we prove that for a cyclic group A ⊂ GL(n, ℂ) of type ${\frac{1}{r}}$(1, …, 1, a) with r coprime to a, A-Hilbert Scheme is smooth and irreducible.

• 대한방사성의약품학회지
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• 제8권2호
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• pp.119-128
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• 2022

2-Dimensional inorganic nanoparticles with high surface area and ion-exchangeable properties have been continuously growing based on nanotechnology in the field of nanomedicine. Among one of the 2-D nanoparticles, layered double hydroxide (LDH) has been intensively explored as drug delivery due to its low toxicity, enhanced cellular permeability, and high drug loading capacity. Moreover, controllable chemical composition makes possible varying isomorphic layered materials for therapy and imaging of diseases. In this review, specific structural characteristics of LDH were introduced, and its potential for application as a biocompatible therapeutic agent and diagnostic one was addressed in nuclear medicine, one of promising fields in nanomedicine.

• Kyungpook Mathematical Journal
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• 제63권1호
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• pp.131-139
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• 2023

Let V_k,n (ℂ) denote the complex Steifel and Gr_k,n (ℂ) the Grassmann manifolds for 1 ≤ k < n. In this paper, we compute, in terms of the Sullivan minimal models, the evaluation subgroups and, more generally, the relative evaluation subgroups of the fibration p : V_k,k+n (ℂ) → Gr_k,k+n (ℂ). In particular, we prove that G_* (Gr_k,k+n (ℂ), V_k,k+n (ℂ) ; p) is isomorphic to G^rel_* (Gr_k,k+n (ℂ), V_k,k+n (ℂ) ; p) ⊕ G_* (V_k,k+n (ℂ)).

• 대한수학회보
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• 제60권2호
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• pp.339-348
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• 2023

It is shown that every nilpotent-invariant module can be decomposed into a direct sum of a quasi-injective module and a square-free module that are relatively injective and orthogonal. This paper is also concerned with rings satisfying every cyclic right R-module is nilpotent-invariant. We prove that R ≅ R₁ × R₂, where R₁, R₂ are rings which satisfy R₁ is a semi-simple Artinian ring and R₂ is square-free as a right R₂-module and all idempotents of R₂ is central. The paper concludes with a structure theorem for cyclic nilpotent-invariant right R-modules. Such a module is shown to have isomorphic simple modules eR and fR, where e, f are orthogonal primitive idempotents such that eRf ≠ 0.

• Kyungpook Mathematical Journal
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• 제63권2호
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• pp.167-173
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• 2023

Let G be a finite group and let ν(G) be the probability that two randomly selected elements of G produce a nilpotent group. In this article we show that for every positive integer n > 0, there is a finite group G such that ${\nu}(G)={\frac{1}{n}}$. We also classify all groups G with ${\nu}(G)={\frac{1}{2}}$. Further, we prove that if G is a solvable nonnilpotent group of even order, then ${\nu}(G){\leq}{\frac{p+3}{4p}}$, where p is the smallest odd prime divisor of |G|, and that equality exists if and only if $\frac{G}{Z_{\infty}(G)}$ is isomorphic to the dihedral group of order 2p where Z_∞(G) is the hypercenter of G. Finally we find an upper bound for ν(G) in terms of |G| where G ranges over all groups of odd square-free order.

• Kyungpook Mathematical Journal
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• 제63권2호
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• pp.199-223
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• 2023

We introduce the constants E[t, X], C_NJ[X] and J[t, X] to describe the asymmetry of the norm. They can be seen as the skew version of the Gao's parameter, von Neumann-Jordan constant and Milman's moduli, respectively. We establish basic properties of these constants, relating them other well known constants, and use these properties to calculate the constants for specific spaces. We then use these constants to study Hilbert spaces, uniformly non-square spaces and their normal structures. With the Banach-Mazur distance, we use them to study isomorphic Banach spaces.

• 대한수학회보
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• 제60권4호
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• pp.1035-1059
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• 2023

Let p be an odd prime. Let E be an elliptic curve defined over a quadratic imaginary field, where p splits completely. Suppose E has supersingular reduction at primes above p. Under appropriate hypotheses, we extend the results of [17] to ℤ²_p-extensions. We define and study the fine double-signed residual Selmer groups in these settings. We prove that for two residually isomorphic elliptic curves, the vanishing of the signed 𝜇-invariants of one elliptic curve implies the vanishing of the signed 𝜇-invariants of the other. Finally, we show that the Pontryagin dual of the Selmer group and the double-signed Selmer groups have no non-trivial pseudo-null submodules for these extensions.

• 대한수학회지
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• 제60권6호
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• pp.1303-1336
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• 2023

In this paper, we study the 𝜂-parallelism of the Ricci operator of almost Kenmotsu 3-manifolds. First, we prove that an almost Kenmotsu 3-manifold M satisfying ∇_𝜉h = -2𝛼h𝜑 for some constant 𝛼 has dominantly 𝜂-parallel Ricci operator if and only if it is locally symmetric. Next, we show that if M is an H-almost Kenmotsu 3-manifold satisfying ∇_𝜉h = -2𝛼h𝜑 for a constant 𝛼, then M is a Kenmotsu 3-manifold or it is locally isomorphic to certain non-unimodular Lie group equipped with a left invariant almost Kenmotsu structure. The dominantly 𝜂-parallelism of the Ricci operator is equivalent to the local symmetry on homogeneous almost Kenmotsu 3-manifolds.