Chan Rhan Huh - Academia.edu (original) (raw)

Papers by Chan Rhan Huh

Journal of the Korean Mathematical Society, 2005

Bulletin of the Korean Mathematical Society, 2008

Let R be a ring and I be a proper ideal of R. For the case of R being commutative, Anderson prove... more Let R be a ring and I be a proper ideal of R. For the case of R being commutative, Anderson proved that (*) there are only finitely many prime ideals minimal over I whenever every prime ideal minimal over I is finitely generated. We in this note extend the class of rings that satisfies the condition (*) to noncommutative rings, so called homomorphically IFP, which is a generalization of commutative rings. As a corollary we obtain that there are only finitely many minimal prime ideals in the polynomial ring over R when every minimal prime ideal of a homomorphically IFP ring R is finitely generated.

East Asian mathematical journal, 2017

It is a well-known fact that a ring R is regular if and only if every left R-modules is flat. In ... more It is a well-known fact that a ring R is regular if and only if every left R-modules is flat. In this article we prove that a ring R is strongly regular if and only if every left R-modules is reduced if and only if every left-R modules is quasi-reduced.

Bulletin of the Korean Mathematical Society, 2002

Bulletin of the Korean Mathematical Society, 2013

We continue the study of McCoy condition to analyze zerodividing polynomials for the constant ann... more We continue the study of McCoy condition to analyze zerodividing polynomials for the constant annihilators in the ideals generated by the coefficients. In the process we introduce the concept of ideal-π-McCoy rings, extending known results related to McCoy condition. It is shown that the class of ideal-π-McCoy rings contains both strongly McCoy rings whose non-regular polynomials are nilpotent and 2-primal rings. We also investigate relations between the ideal-π-McCoy property and other standard ring theoretic properties. Moreover we extend the class of ideal-π-McCoy rings by examining various sorts of ordinary ring extensions.

Bulletin of the Korean Mathematical Society, 2007

Counterexamples on p. p. -rings

Kyungpook Mathematical Journal

A ring R is called right PP if every principal right ideal of R is projective as a right ideal. I... more A ring R is called right PP if every principal right ideal of R is projective as a right ideal. It was shown by S. Jøndrup [Proc. Am. Math. Soc. 28, 431-435 (1971; Zbl 0195.32703)] that if R is a commutative PP ring, then R[x] is a PP ring. It is implicitly well-known that there exists a commutative PP ring R for which the formal power series ring R[[x]] is not PP. In this paper such an example is introduced. Also, G. F. Birkenmeier, J. Y. Kim and the reviewer [in On quasi-Baer rings, Contemp. Math. (to appear)] provide a commutative PP ring R such that R[[x]] is not PP.

A Note on Pi−regular{\Pi}-regularPi−regular Rings

Kyungpook Mathematical Journal

Throughout this paper, all rings are associative rings with identity. The prime radical of a ring... more Throughout this paper, all rings are associative rings with identity. The prime radical of a ring R and the set of nilpotent elements in R are denoted by P(R) and N(R), respectively. In this paper we show that a ring R, in which N(R) forms a 2-sided ideal, is an (S,2)-ring if and only if every idempotent in R is a sum of two units in R, this result may be a generalization of [1, Theorem 4] on such rings and moreover on 2-primal rings. For a commutative ring R, if 2 = 1 + 1 is a unit (i.e. invertible element) in R then R is an (S,2)-ring. As generalizations of commutative rings, there are PI-rings, 2-primal rings and duo rings etc. By Fisher-Snider [5], the preceding argument is also true for PI-rings. As another generalization, Badawi [1] proves that for duo rings the result holds. In this paper we obtain the result, as a corollary of our main result, on a ring R in which N(R) forms a 2-sided ideal (we call such a ring an NI-ring for simplicity in this paper), hiring the method of p...

Journal of Pure and Applied Algebra, 2004

Our study in this note is concentrated on extending the class of strongly-regular rings, observin... more Our study in this note is concentrated on extending the class of strongly-regular rings, observing the structures of them. We call a ring locally ÿnite if every ÿnite subset in it generates a ÿnite semigroup multiplicatively. We ÿrst study the structures of locally ÿnite rings and then study relations between locally ÿnite rings and other related rings. We also study the strong-regularity of some kinds of semiperfect rings with nil Jacobson radicals.

Journal of Pure and Applied Algebra, 2005

Symmetric rings were introduced by Lambek to unify sheaf representations of commutative rings and... more Symmetric rings were introduced by Lambek to unify sheaf representations of commutative rings and reduced rings. We continue the study of symmetric rings, discussing basic examples and extensions. From any given reduced ring we first construct a nonreduced symmetric ring, and observe the form of the minimal noncommutative symmetric ring of order 16 up to isomorphism. We next show that polynomial rings over symmetric rings need not be symmetric and that classical right quotient rings of right Ore symmetric rings are symmetric. We also construct more examples of symmetric rings and counterexamples to several naturally raised situations in the process.

Journal of Pure and Applied Algebra, 2002

This paper concerns two conditions, called right p.p. and generalized right p.p., which are gener... more This paper concerns two conditions, called right p.p. and generalized right p.p., which are generalizations of Baer rings and von Neumann regular rings. We study the subrings and extensions of them, adding proper examples and counterexamples to some situations and questions that occur naturally in the process of this paper.

Armendariz Rings and Semicommutative Rings

Communications in Algebra, 2002

In this note we concern the structures of Armendariz rings and semicommutative rings which are ge... more In this note we concern the structures of Armendariz rings and semicommutative rings which are generalizations of reduced rings, the classical right quotient rings of Armendariz rings, the polynomial rings over semicommutative rings, and the relationships between Armendariz ...

On exchange rings with primitive factor rings artinian

Communications in Algebra, 2000

ABSTRACT W. K. Nicholson [Trans. Am. Math. Soc. 229, 269-278 (1977; Zbl 0352.16006)] has defined ... more ABSTRACT W. K. Nicholson [Trans. Am. Math. Soc. 229, 269-278 (1977; Zbl 0352.16006)] has defined a ring to be clean if each of its elements is the sum of an idempotent and a unit and has proved that every clean ring is an exchange ring. Camillo and Yu have proved that the converse is not true. However, Nicholson has proved that both notions are equivalent for ring in which every idempotent is central. The authors prove that both notions are equivalent for a ring R for which every right primitive factor ring is Artinian. In particular, this result is true for a PI-ring since for a PI-ring every right primitive factor ring is Artinian. They deduce Yu’s result as a special case since for a right quasi-duo ring every right primitive factor ring is Artinian.

On rings in which every maximal one-sided ideal contains a maximal ideal

Communications in Algebra, 1999

Given a ring R, consider the condition: () every maximal right ideal of R contains a maximal idea... more Given a ring R, consider the condition: () every maximal right ideal of R contains a maximal ideal of R. We show that, for a ring R and 0 ≠ e = e ∈ R such that ele ⫋ eRe every proper ideal I of RR satisfies () if and only if eRe satisfies (). Hence with the help of some other results, () is a Morita invariant property. For a simple ring RR[x] satisfies () if and only if R[x] is not right primitive. By this result, if R is a division ring and R[x] satisfies (), then the Jacobson conjecture holds. We also show that for a finite centralizing extension S of a ring RR satisfies () if and only if S satisfies ().

Questions on 2-primal rings

Communications in Algebra, 1998

ABSTRACT A ring is called 2-primal if its prime radical is the set of all nilpotent elements. In ... more ABSTRACT A ring is called 2-primal if its prime radical is the set of all nilpotent elements. In [G. F. Birkenmeier, H. E. Heatherly and E. K. Lee, Ring Theory, Proc. Biennial Ohio State-Denison Conf. 1992, 102-129 (1993; Zbl 0853.16022)], the following questions were raised: Is the direct product of an arbitrary set of 2-primal rings a 2-primal ringΦ Is the formal power series ring over a 2-primal ring a 2-primal ringΦ For the first question, in Armendariz’s example [Example 1.6, G. F. Birkenmeier, H. E. Heatherly and E. K. Lee, in Near-Rings and Near-Fields, Proc. Conf., Fredericton, New Brunswick, Canada 1993, 63-73 (1995; Zbl 0839.16040)], there is a set of non-isomorphic 2-primal rings whose direct product is not 2-primal. By this example, the first question is answered in the negative. The authors answer the second question in the negative by constructing two examples of rings R whose formal power series rings are not 2-primal. In one of them, R is a ring with a polynomial identity. By the help of one of these examples, they also can construct an example which answers the first question negatively. Independently, both questions are answered in the negative by G. Marks [in Advances in Ring Theory, Trends in Mathematics, 239-245 (1997; Zbl 0890.16010)].

Communications in Algebra, 1996

On Rings Whose Strongly Prime Ideals Are Completely Prime

Algebra Colloquium, 2010

Kaplansky introduced the concept of the K-rings, concerning the commutativity of rings. In this p... more Kaplansky introduced the concept of the K-rings, concerning the commutativity of rings. In this paper, we concentrate on a property of K-rings, introducing the concept of the strongly NI rings, which is stronger than NI-ness. We first examine the relations among the concepts concerned with K-rings and strongly NI rings, constructing necessary examples in the process. We also show that strong NI-ness is a hereditary radical property.

Journal of the Association of Korean Geographers, 2019

This article introduces one livelihood Hansalim thought of Jang Il-Soon and suggests a comprehens... more This article introduces one livelihood Hansalim thought of Jang Il-Soon and suggests a comprehensive and ethically conscious understanding of ecosystem integrating natural and religious worldview. A crisis in the ecosystem was first diagnosed as a problem within the ecosystem itself caused by political, economic and social factors. It is now understood that the problem is rooted in the way humans think about nature. The Life Thought of Jang Il-Soon comprises social and spiritual pursuits emphasizing ethical thinking and practice to overcome environmental crisis. One Livelihood Hansalim Movement of Jang Il-Soon integrates Donghak and Christian teachings, and emphasizes that we as human beings have unique rights and obligations, but are a part of all living things where these rights and obligations must be reconsidered through integrated aspects. The main principles of life-centeredness and One Livelihood give us the foundations for ecological awareness and depth of religious mind. These thoughts will allow us to transform ourselves and their practices could be begun in local communities. The Hansalim Movement propose an integrative environmental ethics, and could be used for the base of environmental education toward sustainable future.

Journal of the Korean Mathematical Society, 2005

Bulletin of the Korean Mathematical Society, 2008

Let R be a ring and I be a proper ideal of R. For the case of R being commutative, Anderson prove... more Let R be a ring and I be a proper ideal of R. For the case of R being commutative, Anderson proved that (*) there are only finitely many prime ideals minimal over I whenever every prime ideal minimal over I is finitely generated. We in this note extend the class of rings that satisfies the condition (*) to noncommutative rings, so called homomorphically IFP, which is a generalization of commutative rings. As a corollary we obtain that there are only finitely many minimal prime ideals in the polynomial ring over R when every minimal prime ideal of a homomorphically IFP ring R is finitely generated.

East Asian mathematical journal, 2017

It is a well-known fact that a ring R is regular if and only if every left R-modules is flat. In ... more It is a well-known fact that a ring R is regular if and only if every left R-modules is flat. In this article we prove that a ring R is strongly regular if and only if every left R-modules is reduced if and only if every left-R modules is quasi-reduced.

Bulletin of the Korean Mathematical Society, 2002

Bulletin of the Korean Mathematical Society, 2013

We continue the study of McCoy condition to analyze zerodividing polynomials for the constant ann... more We continue the study of McCoy condition to analyze zerodividing polynomials for the constant annihilators in the ideals generated by the coefficients. In the process we introduce the concept of ideal-π-McCoy rings, extending known results related to McCoy condition. It is shown that the class of ideal-π-McCoy rings contains both strongly McCoy rings whose non-regular polynomials are nilpotent and 2-primal rings. We also investigate relations between the ideal-π-McCoy property and other standard ring theoretic properties. Moreover we extend the class of ideal-π-McCoy rings by examining various sorts of ordinary ring extensions.

Bulletin of the Korean Mathematical Society, 2007

Counterexamples on p. p. -rings

Kyungpook Mathematical Journal

A ring R is called right PP if every principal right ideal of R is projective as a right ideal. I... more A ring R is called right PP if every principal right ideal of R is projective as a right ideal. It was shown by S. Jøndrup [Proc. Am. Math. Soc. 28, 431-435 (1971; Zbl 0195.32703)] that if R is a commutative PP ring, then R[x] is a PP ring. It is implicitly well-known that there exists a commutative PP ring R for which the formal power series ring R[[x]] is not PP. In this paper such an example is introduced. Also, G. F. Birkenmeier, J. Y. Kim and the reviewer [in On quasi-Baer rings, Contemp. Math. (to appear)] provide a commutative PP ring R such that R[[x]] is not PP.

A Note on Pi−regular{\Pi}-regularPi−regular Rings

Kyungpook Mathematical Journal

Throughout this paper, all rings are associative rings with identity. The prime radical of a ring... more Throughout this paper, all rings are associative rings with identity. The prime radical of a ring R and the set of nilpotent elements in R are denoted by P(R) and N(R), respectively. In this paper we show that a ring R, in which N(R) forms a 2-sided ideal, is an (S,2)-ring if and only if every idempotent in R is a sum of two units in R, this result may be a generalization of [1, Theorem 4] on such rings and moreover on 2-primal rings. For a commutative ring R, if 2 = 1 + 1 is a unit (i.e. invertible element) in R then R is an (S,2)-ring. As generalizations of commutative rings, there are PI-rings, 2-primal rings and duo rings etc. By Fisher-Snider [5], the preceding argument is also true for PI-rings. As another generalization, Badawi [1] proves that for duo rings the result holds. In this paper we obtain the result, as a corollary of our main result, on a ring R in which N(R) forms a 2-sided ideal (we call such a ring an NI-ring for simplicity in this paper), hiring the method of p...

Journal of Pure and Applied Algebra, 2004

Our study in this note is concentrated on extending the class of strongly-regular rings, observin... more Our study in this note is concentrated on extending the class of strongly-regular rings, observing the structures of them. We call a ring locally ÿnite if every ÿnite subset in it generates a ÿnite semigroup multiplicatively. We ÿrst study the structures of locally ÿnite rings and then study relations between locally ÿnite rings and other related rings. We also study the strong-regularity of some kinds of semiperfect rings with nil Jacobson radicals.

Journal of Pure and Applied Algebra, 2005

Symmetric rings were introduced by Lambek to unify sheaf representations of commutative rings and... more Symmetric rings were introduced by Lambek to unify sheaf representations of commutative rings and reduced rings. We continue the study of symmetric rings, discussing basic examples and extensions. From any given reduced ring we first construct a nonreduced symmetric ring, and observe the form of the minimal noncommutative symmetric ring of order 16 up to isomorphism. We next show that polynomial rings over symmetric rings need not be symmetric and that classical right quotient rings of right Ore symmetric rings are symmetric. We also construct more examples of symmetric rings and counterexamples to several naturally raised situations in the process.

Journal of Pure and Applied Algebra, 2002

This paper concerns two conditions, called right p.p. and generalized right p.p., which are gener... more This paper concerns two conditions, called right p.p. and generalized right p.p., which are generalizations of Baer rings and von Neumann regular rings. We study the subrings and extensions of them, adding proper examples and counterexamples to some situations and questions that occur naturally in the process of this paper.

Armendariz Rings and Semicommutative Rings

Communications in Algebra, 2002

In this note we concern the structures of Armendariz rings and semicommutative rings which are ge... more In this note we concern the structures of Armendariz rings and semicommutative rings which are generalizations of reduced rings, the classical right quotient rings of Armendariz rings, the polynomial rings over semicommutative rings, and the relationships between Armendariz ...

On exchange rings with primitive factor rings artinian

Communications in Algebra, 2000

ABSTRACT W. K. Nicholson [Trans. Am. Math. Soc. 229, 269-278 (1977; Zbl 0352.16006)] has defined ... more ABSTRACT W. K. Nicholson [Trans. Am. Math. Soc. 229, 269-278 (1977; Zbl 0352.16006)] has defined a ring to be clean if each of its elements is the sum of an idempotent and a unit and has proved that every clean ring is an exchange ring. Camillo and Yu have proved that the converse is not true. However, Nicholson has proved that both notions are equivalent for ring in which every idempotent is central. The authors prove that both notions are equivalent for a ring R for which every right primitive factor ring is Artinian. In particular, this result is true for a PI-ring since for a PI-ring every right primitive factor ring is Artinian. They deduce Yu’s result as a special case since for a right quasi-duo ring every right primitive factor ring is Artinian.

On rings in which every maximal one-sided ideal contains a maximal ideal

Communications in Algebra, 1999

Given a ring R, consider the condition: () every maximal right ideal of R contains a maximal idea... more Given a ring R, consider the condition: () every maximal right ideal of R contains a maximal ideal of R. We show that, for a ring R and 0 ≠ e = e ∈ R such that ele ⫋ eRe every proper ideal I of RR satisfies () if and only if eRe satisfies (). Hence with the help of some other results, () is a Morita invariant property. For a simple ring RR[x] satisfies () if and only if R[x] is not right primitive. By this result, if R is a division ring and R[x] satisfies (), then the Jacobson conjecture holds. We also show that for a finite centralizing extension S of a ring RR satisfies () if and only if S satisfies ().

Questions on 2-primal rings

Communications in Algebra, 1998

ABSTRACT A ring is called 2-primal if its prime radical is the set of all nilpotent elements. In ... more ABSTRACT A ring is called 2-primal if its prime radical is the set of all nilpotent elements. In [G. F. Birkenmeier, H. E. Heatherly and E. K. Lee, Ring Theory, Proc. Biennial Ohio State-Denison Conf. 1992, 102-129 (1993; Zbl 0853.16022)], the following questions were raised: Is the direct product of an arbitrary set of 2-primal rings a 2-primal ringΦ Is the formal power series ring over a 2-primal ring a 2-primal ringΦ For the first question, in Armendariz’s example [Example 1.6, G. F. Birkenmeier, H. E. Heatherly and E. K. Lee, in Near-Rings and Near-Fields, Proc. Conf., Fredericton, New Brunswick, Canada 1993, 63-73 (1995; Zbl 0839.16040)], there is a set of non-isomorphic 2-primal rings whose direct product is not 2-primal. By this example, the first question is answered in the negative. The authors answer the second question in the negative by constructing two examples of rings R whose formal power series rings are not 2-primal. In one of them, R is a ring with a polynomial identity. By the help of one of these examples, they also can construct an example which answers the first question negatively. Independently, both questions are answered in the negative by G. Marks [in Advances in Ring Theory, Trends in Mathematics, 239-245 (1997; Zbl 0890.16010)].

Communications in Algebra, 1996

On Rings Whose Strongly Prime Ideals Are Completely Prime

Algebra Colloquium, 2010

Kaplansky introduced the concept of the K-rings, concerning the commutativity of rings. In this p... more Kaplansky introduced the concept of the K-rings, concerning the commutativity of rings. In this paper, we concentrate on a property of K-rings, introducing the concept of the strongly NI rings, which is stronger than NI-ness. We first examine the relations among the concepts concerned with K-rings and strongly NI rings, constructing necessary examples in the process. We also show that strong NI-ness is a hereditary radical property.

Journal of the Association of Korean Geographers, 2019

This article introduces one livelihood Hansalim thought of Jang Il-Soon and suggests a comprehens... more This article introduces one livelihood Hansalim thought of Jang Il-Soon and suggests a comprehensive and ethically conscious understanding of ecosystem integrating natural and religious worldview. A crisis in the ecosystem was first diagnosed as a problem within the ecosystem itself caused by political, economic and social factors. It is now understood that the problem is rooted in the way humans think about nature. The Life Thought of Jang Il-Soon comprises social and spiritual pursuits emphasizing ethical thinking and practice to overcome environmental crisis. One Livelihood Hansalim Movement of Jang Il-Soon integrates Donghak and Christian teachings, and emphasizes that we as human beings have unique rights and obligations, but are a part of all living things where these rights and obligations must be reconsidered through integrated aspects. The main principles of life-centeredness and One Livelihood give us the foundations for ecological awareness and depth of religious mind. These thoughts will allow us to transform ourselves and their practices could be begun in local communities. The Hansalim Movement propose an integrative environmental ethics, and could be used for the base of environmental education toward sustainable future.