• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.779-791
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• 2016

Transfer of homological properties under base change is a classical field of study. Let $R{\rightarrow}S$ be a ring homomorphism. The relations of Gorenstein projective (or Gorenstein injective) dimensions of unbounded complexes between $U{\otimes}^L_RX$(or $RHom_R(X,U)$) and X are considered, where X is an R-complex and U is an S-complex. In addition, some sufficient conditions are given under which the equalities $G-dim_S(U{\otimes}^L_RX)=G-dim_RX+pd_SU$ and $Gid_S(RHom_R(X,U))=G-dim_RX+id_SU$ hold.

• Communications of the Korean Mathematical Society
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• v.29 no.1
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• pp.27-36
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• 2014

In this paper, we introduce the notion of f-derivations from a semilattice S to a lattice L, as a generalization of derivation and f-derivation of lattices. Also, we define the simple f-derivation from S to L, and research the properties of them and the conditions for a lattice L to be distributive. Finally, we prove that a distributive lattice L is isomorphic to the class $SD_f(S,L)$ of all simple f-derivations on S to L for every ${\wedge}$-homomorphism $f:S{\rightarrow}L$ such that $f(x_0){\vee}f(y_0)=1$ for some $x_0,y_0{\in}S$, in particular, $$L{\simeq_-}=SD_f(S,L)$$ for every ${\wedge}$-homomorphism $f:S{\rightarrow}L$ such that $f(x_0)=1$ for some $x_0{\in}S$.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.319-329
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• 2017

Let I denote an ideal of a Noetherian local ring (R, m). Let M denote a finitely generated R-module. We study the endomorphism ring of the local cohomology module $H^c_I(M)$, c = grade(I, M). In particular there is a natural homomorphism $$Hom_{\hat{R}^I}({\hat{M}}^I,\;{\hat{M}}^I){\rightarrow}Hom_R(H^c_I(M),\;H^c_I(M))$$, $where{\hat{\cdot}}^I$ denotes the I-adic completion functor. We provide sufficient conditions such that it becomes an isomorphism. Moreover, we study a homomorphism of two such endomorphism rings of local cohomology modules for two ideals $J{\subset}I$ with the property grade(I, M) = grade(J, M). Our results extends constructions known in the case of M = R (see e.g. [8], [17], [18]).

• East Asian mathematical journal
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• v.18 no.2
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• pp.195-203
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• 2002

Let R be a prime ring with characteristic not two and U is a nonzero left ideal of R which contains no nonzero nilpotent right ideal as a ring. For a $(\sigma,\tau)$-derivation d : R$\rightarrow$R, we prove the following results: (1) If d is an endomorphism on R then d=0. (2) If d is an anti-endomorphism on R then d=0. (3) If d(xy)=d(yx), for all x, y$\in$R then R is commutative. (4) If d is an homomorphism or anti-homomorphism on U then d=0.

• Kyungpook Mathematical Journal
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• v.50 no.2
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• pp.329-336
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• 2010

In this paper, we introduce a partitioning subsemimodule of a semimodule over a semiring which is useful to develop the quotient structure of semimodule. Indeed we prove : 1) The quotient semimodule M=N(Q) is essentially independent of choice of Q. 2) If f : M ${\rightarrow}$ M' is a maximal R-semimodule homomorphism, then $M/kerf_{(Q)}\;\cong\;M'$. 3) Every partitioning subsemimodule is subtractive. 4) Let N be a Q-subsemimodule of an R-semimodule M. Then A is a subtractive subsemimodule of M with $N{\subseteq}A$ if and only if $A/N_{(Q{\cap}A)}\;=\;\{q+N:q{\in}Q{\cap}A\}$ is a subtractive subsemimodule of $M/N_{(Q)}$.

• Kyungpook Mathematical Journal
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• v.51 no.1
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• pp.69-76
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• 2011

In this paper, we introduce a partitioning ideal of a ternary semiring which is useful to develop the quotient structure of ternary semiring. Indeed we prove : 1) The quotient ternary semiring S/$I_{(Q)}$ is essentially independent of choice of Q. 2) If f : S ${\rightarrow}$ S' is a maximal ternary semiring homomorphism, then S/ker $f_{(Q)}$ ${\cong}$ S'. 3) Every partitioning ideal is subtractive. 4) Let I be a Q-ideal of a ternary semiring S. Then A is a subtractive ideal of S with I ${\subseteq}$ A if and only if A/$I_{(Q{\cap}A)}$ = {q + I : q ${\in}$ Q ${\cap}$ A} is a subtractive idea of S/$I_{(Q)}$.

• Kyungpook Mathematical Journal
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• v.48 no.4
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• pp.529-543
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• 2008

It is shown that every almost linear mapping h : $A{\rightarrow}B$ of a unital PoissonC*-algebra A to a unital Poisson C*-algebra B is a Poisson C*-algebra homomorph when $h(2^nuy)\;=\;h(2^nu)h(y)$ or $h(3^nuy)\;=\;h(3^nu)h(y)$ for all $y\;\in\;A$, all unitary elements $u\;\in\;A$ and n = 0, 1, 2,$\codts$, and that every almost linear almost multiplicative mapping h : $A{\rightarrow}B$ is a Poisson C*-algebra homomorphism when h(2x) = 2h(x) or h(3x) = 3h(x for all $x\;\in\;A$. Here the numbers 2, 3 depend on the functional equations given in the almost linear mappings or in the almost linear almost multiplicative mappings. We prove the Cauchy-Rassias stability of Poisson C*-algebra homomorphisms in unital Poisson C*-algebras, and of homomorphisms in Poisson Banach modules over a unital Poisson C*-algebra.

• The Journal of Natural Sciences
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• v.11 no.1
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• pp.11-14
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• 1999

Let G be a compact Lie group and let $\rho$ : G $\rightarrow O(2) be a homomorphism. Denote by V the G-module associated with $\rho$ and by S(V) the unit circle of V. In this paper, we show that if G is abelian, then a real G-vector bundle over S(V) is isomorphic to Whitney sum of real G-line or G-plane bundles.

• Communications of the Korean Mathematical Society
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• v.36 no.1
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• pp.1-10
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• 2021

Let R and S be two commutative rings, J be an ideal of S and f : R → S be a ring homomorphism. The amalgamation of R and S along J with respect to f, denoted by R ⋈f J, is the special subring of R × S defined by R ⋈f J = {(a, f(a) + j) | a ∈ R, j ∈ J}. In this paper, we study some basic properties of a special kind of R ⋈f J-modules, called the amalgamation of M and N along J with respect to ��, and defined by M ⋈�� JN := {(m, ��(m) + n) | m ∈ M and n ∈ JN}, where �� : M → N is an R-module homomorphism. The new results generalize some known results on the amalgamation of rings and the duplication of a module along an ideal.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.725-743
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• 2022

In this paper, the concepts of ω-linked homomorphisms, the ω_𝜙-operation, and DW_𝜙 rings are introduced. Also the relationships between ω_𝜙-ideals and ω-ideals over a ω-linked homomorphism 𝜙 : R → T are discussed. More precisely, it is shown that every ω_𝜙-ideal of T is a ω-ideal of T. Besides, it is shown that if T is not a DW_𝜙 ring, then T must have an infinite number of maximal ω_𝜙-ideals. Finally we give an application of Cohen's Theorem over ω-factor rings, namely it is shown that an integral domain R is an SM-domain with ω-dim(R) ≤ 1, if and only if for any nonzero ω-ideal I of R, (R/I)_ω is an Artinian ring, if and only if for any nonzero element α ∈ R, (R/(a))_ω is an Artinian ring, if and only if for any nonzero element α ∈ R, R satisfies the descending chain condition on ω-ideals of R containing a.