• 대한수학회보
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• 제47권5호
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• pp.907-913
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• 2010

Let D be an integrally closed domain with quotient field K, * be a star operation on D, X, Y be indeterminates over D, $N_*\;=\;\{f\;{\in}\;D[X]|\;(c_D(f))^*\;=\;D\}$ and $R\;=\;D[X]_{N_*}$. Let b be the b-operation on R, and let $*_c$ be the star operation on D defined by $I^{*_c}\;=\;(ID[X]_{N_*})^b\;{\cap}\;K$. Finally, let Kr(R, b) (resp., Kr(D, $*_c$)) be the Kronecker function ring of R (resp., D) with respect to Y (resp., X, Y). In this paper, we show that Kr(R, b) $\subseteq$ Kr(D, $*_c$) and Kr(R, b) is a kfr with respect to K(Y) and X in the notion of [2]. We also prove that Kr(R, b) = Kr(D, $*_c$) if and only if D is a $P{\ast}MD$. As a corollary, we have that if D is not a $P{\ast}MD$, then Kr(R, b) is an example of a kfr with respect to K(Y) and X but not a Kronecker function ring with respect to K(Y) and X.

• 대한수학회보
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• 제46권5호
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• pp.1013-1018
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• 2009

Let * be an e.a.b. star operation on an integrally closed domain D, and let $K\gamma$(D, *) be the Kronecker function ring of D. We show that if D is a P*MD, then the mapping $D_{\alpha}{\mapsto}K{\gamma}(D_{\alpha},\;{\upsilon})$ is a bijection from the set {$D_{\alpha}$} of *-linked overrings of D into the set of overrings of $K{\gamma}(D,\;{\upsilon})$. This is a generalization of [5, Proposition 32.19] that if D is a Pr$\ddot{u}$fer domain, then the mapping $D_{\alpha}{\mapsto}K_{\gamma}(D_{\alpha},\;b)$ is a one-to-one mapping from the set {$D_{\alpha}$} of overrings of D onto the set of overrings of $K_{\gamma}$(D, b).

• 대한수학회지
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• 제50권2호
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• pp.393-410
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• 2013

Let $R{\subseteq}T$ be an extension of integral domains, X be an indeterminate over T, and R[X] and T[X] be polynomial rings. Then $R{\subseteq}T$ is said to be LCM-stable if $(aR{\cap}bR)T=aT{\cap}bT$ for all $0{\neq}a,b{\in}R$. Let $w_A$ be the so-called $w$-operation on an integral domain A. In this paper, we introduce the notions of $w(e)$- and $w$-LCM-stable extensions: (i) $R{\subseteq}T$ is $w(e)$-LCM-stable if $((aR{\cap}bR)T)_{w_T}=aT{\cap}bT$ for all $0{\neq}a,b{\in}R$ and (ii) $R{\subseteq}T$ is $w$-LCM-stable if $((aR{\cap}bR)T)_{w_R}=(aT{\cap}bT)_{w_R}$ for all $0{\neq}a,b{\in}R$. We prove that LCM-stable extensions are both $w(e)$-LCM-stable and $w$-LCM-stable. We also generalize some results on LCM-stable extensions. Among other things, we show that if R is a Krull domain (resp., $P{\upsilon}MD$), then $R{\subseteq}T$ is $w(e)$-LCM-stable (resp., $w$-LCM-stable) if and only if $R[X]{\subseteq}T[X]$ is $w(e)$-LCM-stable (resp., $w$-LCM-stable).

• Journal of applied mathematics & informatics
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• 제41권3호
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• pp.615-631
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• 2023

A power cordial labeling of a graph G = (V (G), E(G)) is a bijection f : V (G) → {1, 2, ..., |V (G)|} such that an edge e = uv is assigned the label 1 if f(u) = (f(v))ⁿ or f(v) = (f(u))ⁿ, For some n ∈ ℕ ∪ {0} and the label 0 otherwise, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. In this paper, we investigate power cordial labeling for helm graph, flower graph, gear graph, fan graph and jewel graph as well as larger graphs obtained from star and bistar using graph operations.