• Journal of the Korean Mathematical Society
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• v.51 no.2
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• pp.325-344
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• 2014

The aim of this paper is to generalize the theory of Hopf-Ore extension on Hopf algebras to Hopf group coalgebras. First the concept of Hopf-Ore extension of Hopf group coalgebra is introduced. Then we will give the necessary and sufficient condition for the Ore extensions to become a Hopf group coalgebra, and certain isomorphism between Ore extensions of Hopf group coalgebras are discussed.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.2
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• pp.191-193
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• 2019

It is shown that a left Ore extension $D[x;{\sigma}]$ over a division ring D is a left Goldie ring with no zero-divisor and that its left Goldie quotient is an injective hull of $D[x;{\sigma}]$.

• The Journal of the Petrological Society of Korea
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• v.21 no.4
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• pp.397-403
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• 2012

Fe-REE deposits occurred in Jaeunri, Hongcheon formed by carbonatitic melts consist of 3 parts such as northern, middle and southern ore bodies showing discontinuous distribution, and extension shape of ore bodies can be figured through field survey and geometric analysis. Foliations in gneiss around northern and middle ore bodies represent NNE, whereas toward south its direction changes gradually from NE to ENE and finally N-S direction appears in southern ore body. From Jaeungyo to Saemaeulgyo geometric analysis from field work gives that fold shape in this area is open synclinal fold concavely and gently to NW with $45^{\circ}$ northwestward plunging axis, in contrast small scale anticline with $45^{\circ}$ northwestward plunging axis in Yagsooteo area near western part of Saemaeulgyo. Dragging effect could be occurred on these folds by WNW trending dextral strike-slip fault from Yagsooteo to Saemaeulgyo. New ore body can be confirmed from folding structure estimated by trend of foliation, and thus unidentified ore body may be exist under alluvial surface from middle to southern ore body and its distribution could show reversed "ㄱ", one of Korean consonants, considering with estimated strike and dip of foliations. This estimated extension of ore body figured out by structural analysis in the studied area works an important role for measuring of ore reserve and selecting of drilling site to find new ore body.

• Kyungpook Mathematical Journal
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• v.59 no.4
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• pp.603-615
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• 2019

Let R be an associative unital ring with an endomorphism α and α-derivation δ. The constant products of elements in Ore extension rings, when the coefficient ring is reversible, is investigated. We show that if f(x) = ∑ⁿ_i=0 a_ixⁱ and g(x) = ∑^m_j=0 b_jx^j be nonzero elements in Ore extension ring R[x; α, δ] such that g(x)f(x) = c ∈ R, then there exist non-zero elements r, a ∈ R such that rf(x) = ac, when R is an (α, δ)-compatible ring which is reversible. Among applications, we give an exact characterization of the unit elements in R[x; α, δ], when the coeficient ring R is (α, δ)-compatible. Furthermore, it is shown that if R is a weakly 2-primal ring which is (α, δ)-compatible, then J(R[x; α, δ]) = N iℓ(R)[x; α, δ]. Some other applications and examples of rings with this property are given, with an emphasis on certain classes of NI rings. As a consequence we obtain generalizations of the many results in the literature. As the final part of the paper we construct examples of rings that explain the limitations of the results obtained and support our main results.

• Communications of the Korean Mathematical Society
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• v.21 no.1
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• pp.1-9
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• 2006

A ring R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. Let R be a ring, G be an ordered monoid acting on R by $\beta$ and R be G-compatible. It is shown that R is (left principally) quasi-Baer if and only if skew monoid ring $R_{\beta}[G]$ is (left principally) quasi-Baer. If G is an abelian monoid, then R is (left principally) quasi-Baer if and only if the Cohn-Jordan extension $A(R,\;\beta)$ is (left principally) quasi-Baer if and only if left Ore quotient ring $G^{-1}R_{\beta}[G]$ is (left principally) quasi-Baer.

• Proceedings of the KSEEG Conference
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• 2007.04a
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• pp.21-22
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• 2007

See Full Text

• Economic and Environmental Geology
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• v.11 no.4
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• pp.143-167
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• 1978

The Sangdong tungsten (mostly scheelite) mine is located on the southern limb of a major syncline, the Hambaeg syncline, in a thick sequence of lower Paleozoic sedimentary rocks in the mideastern part of south Korea. Productive scheelite mineralization in Sangdong area is confined to one single formation, the Myobong Slate. Four major ore beds, which have an lateral extension over than 1 km and were not heavily subjected to spatial disturbance, are developed in the Myobong Formation. The original materials of the ore-comprising horizones were probably of either calcareous or silceous sediments. The four ore beds, especially in the case of Main ore bed, display both lateral and vertical zoning. Association quartz-mica-scheelite is predominant in the central, while association hornblende-quartz-diopside-scheelite, diopside-garnet and wollastonite-garnet are developed in this order towards the periphery of the ore beds. Petrologically, two phases of thermometamorphism are recognized. The first phase is represented by the association wollastonite-garnet and diopside-garnet, while the second phase by the association hornblende-quartz-diopside-scheelite and quartz-mica-scheelite. The associations of the second phase do constitute prodctive ore. The high background value of tungsten in the area surrounding the Sangdong mine reveals that the area can be considered a geochemical zone enriched in tungsten. Studies on the trace element patterns were carried out to draw useful criteria for the purpose of future geochemical exploration in the area. The increasing trend of the ratio Rb $({\times}1000)/K_2O$ of the Myobong Slate towards the known mineralization area proved to be indicative for the presence of tungsten mineralization.

• East Asian mathematical journal
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• v.18 no.2
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• pp.271-282
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• 2002

Let R be a ring with an endomorphism $\sigma$ and a derivation $\delta$. An ideal I of R is ($\sigma,\;\delta$)-ideal of R if $\sigma(I){\subseteq}I$ and $\delta(I){\subseteq}I$. An ideal P of R is a ($\sigma,\;\delta$)-prime ideal of R if P(${\neq}R$) is a ($\sigma,\;\delta$)-ideal and for ($\sigma,\;\delta$)-ideals I and J of R, $IJ{\subseteq}P$ implies that $I{\subseteq}P$ or $J{\subseteq}P$. An ideal Q of R is ($\sigma,\;\delta$)-semiprime ideal of R if Q is a ($\sigma,\;\delta$)-ideal and for ($\sigma,\;\delta$)-ideal I of R, $I^2{\subseteq}Q$ implies that $I{\subseteq}Q$. The ($\sigma,\;\delta$)-prime radical (resp. prime radical) is defined by the intersection of all ($\sigma,\;\delta$)-prime ideals (resp. prime ideals) of R and is denoted by $P_{(\sigma,\delta)}(R)$(resp. P(R)). In this paper, the following results are obtained: (1) $P_{(\sigma,\delta)}(R)$ is the smallest ($\sigma,\;\delta$)-semiprime ideal of R; (2) For every extended endomorphism $\bar{\sigma}$ of $\sigma$, the $\bar{\sigma}$-prime radical of an Ore extension $P(R[x;\sigma,\delta])$ is equal to $P_{\sigma,\delta}(R)[x;\sigma,\delta]$.

• East Asian mathematical journal
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• v.38 no.1
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• pp.1-12
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• 2022

Let R be a ring with an endomorphism σ and a σ-derivation δ. R is called (σ, δ)-Baer (resp. (σ, δ)-quasi-Baer, (σ, δ)-p.q.-Baer, (σ, δ)-p.p.) if the right annihilator of every right (σ, δ)-set (resp., (σ, δ)-ideal, principal (σ, δ)-ideal, (σ, δ)-element) of R is generated by an idempotent of R. In this paper, for a given Ore extension A = R[x; σ, δ] of R, the following properties are investigated: If R is a σ-rigid ring in which σ and δ commute, then (1) R is (σ, δ)-Baer if and only if R is (σ, δ)-quasi-Baer if and only if A is (${\bar{\sigma}},\;{\bar{\delta}}$)-Baer if and only if A is (${\bar{\sigma}},\;{\bar{\delta}}$)-quasi-Baer; (2) R is (σ, δ)-p.p. if and only if R is (σ, δ)-p.q.-Baer if and only if A is (${\bar{\sigma}},\;{\bar{\delta}}$)-p.p. if and only if A is (${\bar{\sigma}},\;{\bar{\delta}}$)-p.q.-Baer.

• Korean Journal of Mathematics
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• v.21 no.1
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• pp.41-53
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• 2013

Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, introducing the notion of nil-Armendariz rings. Hizem extended the nil-Armendariz property for polynomial rings onto power-series rings, say nil power-serieswise rings. In this paper, we introduce the notion of power-serieswise CN rings that is a generalization of nil power-serieswise Armendariz rings. Finally, we study the nil-Armendariz property for Ore extensions and skew power series rings.