• Journal of the Chungcheong Mathematical Society
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• v.32 no.1
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• pp.87-95
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• 2019

The homothety classes of lattices in a two dimensional vector space over a nonarchimedean local field form a regular tree ${\mathcal{T}}$ of degree q + 1 on which the projective linear group acts naturally where q is the order of the residue field. We show that for any finite regular combinatorial graph of even degree q + 1, there exists a torsion free discrete subgroup ${\Gamma}$ of the projective linear group such that ${\mathcal{T}}/{\Gamma}$ is isomorphic to the graph.

• Kyungpook Mathematical Journal
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• v.49 no.3
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• pp.419-424
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• 2009

Let J be the Jacobian variety of a hyperelliptic curve over $\mathbb{Q}$. Let M be the field generated by all square roots of rational integers over a finite number field K. Then we prove that the Mordell-Weil group J(M) is the direct sum of a finite torsion group and a free $\mathbb{Z}$-module of infinite rank. In particular, J(M) is not a divisible group. On the other hand, if $\widetilde{M}$ is an extension of M which contains all the torsion points of J over $\widetilde{\mathbb{Q}}$, then $J(\widetilde{M}^{sol})/J(\widetilde{M}^{sol})_{tors}$ is a divisible group of infinite rank, where $\widetilde{M}^{sol}$ is the maximal solvable extension of $\widetilde{M}$.

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.659-668
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• 2021

Let R be a semiprime ring with maximal right ring of quotients Q_mr(R), and let n₁, n₂, …, n_k be k fixed positive integers. Suppose that R is (n₁+n₂+⋯+n_k)!-torsion free, and that f : 𝜌 → Q_mr(R) is an additive map, where 𝜌 is a nonzero right ideal of R. It is proved that if [[…[f(x), x^n₁], …], x^n_k] = 0 for all x ∈ 𝜌, then [f(x), x] = 0 for all x ∈ 𝜌. This gives the result of Beidar et al. [2] for semiprime rings. Moreover, it is also proved that if R is p-torsion, where p is a prime integer with p = Σ^k_i=1 n_i and if f : R → Q_mr(R) is an additive map satisfying [[…[f(x), x^n₁], …], x^n_k] = 0 for all x ∈ R, then [f(x), x] = 0 for all x ∈ R.

• Journal of the Korean Mathematical Society
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• v.32 no.1
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• pp.51-60
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• 1995

Let $(M,g_M,F)$ be a (p+q)-dimensional connected Riemannian manifold with a foliation $F$ of codimension q and a complete bundle-like metric $g_M$ with respect to $F$. Let $Ric_D$ be the transverse Ricci field of $F$ with respect to the transverse Riemannian connection D which is a torsion-free and $g_Q$-metrical connection on the normal bundle Q of $F$. We consider transverse confomal (or, projective) fields of $F$. It is clear that a tranverse Killing field s of $F$ preserves the transverse Ricci field of $F$, that is, $\Theta(s)Ric_D = 0$, where $\Theta(s)$ denotes the transverse Lie differentiation with respect to s.

• Honam Mathematical Journal
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• v.22 no.1
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• pp.9-16
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• 2000

An A-module M is catenary if for each pair of prime submodules K and L of M with $K{\subset}L$ all saturated chains of prime submodules of M from K to L have a common finite length. We show that when A is a Noetherian domain, then every finitely generated A-module is catenary if and only if A is a Dedekind domain or a field. Moreover, a torsion-free divisible A-module M is catenary if and only if the vector space M over Q(A) (the field of fractions of A) is finite dimensional.

• Kyungpook Mathematical Journal
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• v.57 no.3
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• pp.419-431
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• 2017

In this paper, we define a new kind of formal power series rings by using Gaussian binomial coefficients and investigate some properties. More precisely, we call such a ring a Gaussian series ring and study McCoy's theorem, Hermite properties and Noetherian properties.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1905-1914
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• 2013

Let D be an integral domain with quotient field K, M a torsion-free D-module, X an indeterminate, and $N_v=\{f{\in}D[X]|c(f)_v=D\}$. Let $q(M)=M{\otimes}_D\;K$ and $M_{w_D}$={$x{\in}q(M)|xJ{\subseteq}M$ for a nonzero finitely generated ideal J of D with $J_v$ = D}. In this paper, we show that $M_{w_D}=M[X]_{N_v}{\cap}q(M)$ and $(M[X])_{w_{D[X]}}{\cap}q(M)[X]=M_{w_D}[X]=M[X]_{N_v}{\cap}q(M)[X]$. Using these results, we prove that M is a strong Mori D-module if and only if M[X] is a strong Mori D[X]-module if and only if $M[X]_{N_v}$ is a Noetherian $D[X]_{N_v}$-module. This is a generalization of the fact that D is a strong Mori domain if and only if D[X] is a strong Mori domain if and only if $D[X]_{N_v}$ is a Noetherian domain.

• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.885-898
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• 2012

Let m, n, r be nonzero fixed positive integers, R a 2-torsion free prime ring, Q its right Martindale quotient ring, and L a non-central Lie ideal of R. Let D : $R{\rightarrow}R$ be a skew derivation of R and $E(x)=D(x^{m+n+r})-D(x^m)x^{n+r}-x^mD(x^n)x^r-x^{m+n}D(x^r)$. We prove that if $E(x)=0$ for all $x{\in}L$, then D is a usual derivation of R or R satisfies $s_4(x_1,{\ldots},x_4)$, the standard identity of degree 4.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.397-408
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• 2016

Let R be a 2-torsion free prime ring with center Z, U be the Utumi quotient ring, Q be the Martindale quotient ring of R, d be a derivation of R and L be a Lie ideal of R. If $d(uv)^n=d(u)^md(v)^l$ or $d(uv)^n=d(v)^ld(u)^m$ for all $u,v{\in}L$, where m, n, l are xed positive integers, then $L{\subseteq}Z$. We also examine the case when R is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on non-commutative Banach algebras. This result simultaneously generalizes a number of results in the literature.