• Korean Journal of Mathematics
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• v.30 no.3
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• pp.513-524
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• 2022

In this paper, we introduce the notion of tensor product in groups and prove its existence and uniqueness. Next, we provide the Isbell's zigzag theorem for dominions in commutative groups. We then show that in the category of commutative groups dominions are trivial. This enables us to deduce a well known result epis are surjective in the category of commutative groups.

• Kyungpook Mathematical Journal
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• v.46 no.2
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• pp.231-233
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• 2006

We characterize those commutative semigroups S such that every non-isomorphic homomorphic image of S has smaller cardinality than S. We also characterize commutative groups with the same property.

• Honam Mathematical Journal
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• v.45 no.3
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• pp.410-417
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• 2023

Suppose that there are 8 abelian varieties defined over a number field K which satisfy a commutative diagram. We show that if we know that three out of four short exact sequences satisfy the rate formula of Tate-Shafarevich groups, then the unknown short exact sequence satisfies the rate formula of Tate-Shafarevich groups, too.

• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.239-246
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• 2015

If for every elements x and y of an associative algebraic structure (S, ${\cdot}$) there exists a positive integer r such that $ab=b^ra$, then S is called quasi-commutative. Evidently, every abelian group or commutative semigroup is quasi-commutative. Also every finite Hamiltonian group that may be considered as a semigroup, is quasi-commutative however, there are quasi-commutative semigroups which are non-group and non commutative. In this paper, we provide three finitely presented non-commutative semigroups which are quasi-commutative. These are the first given concrete examples of finite semigroups of this type.

• Kyungpook Mathematical Journal
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• v.46 no.2
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• pp.285-295
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• 2006

We present a concept of negative definite functions on a commutative normal hypercomplex system $L_1$(Q, $m$) with basis unity. Negative definite functions were studied in [5] and [4] for commutative groups and semigroups respectively. The definition of such functions on Q is a natural generalization of that defined on a commutative hypergroups.

• Bulletin of the Korean Mathematical Society
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• v.51 no.6
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• pp.1689-1696
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• 2014

Recently, it was proved that every p-Schur ring over an abelian group of order $p^3$ is Schurian. In this paper, we prove that every commutative p-Schur ring over a non-abelian group of order $p^3$ is Schurian.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1549-1557
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• 2015

Characterizations of closed subgroups in abelian groups have been generalized to modules in essentially dierent ways; they are in general inequivalent. Here we consider the relations between these generalizations over commutative rings, and we characterize the commutative rings over which they coincide. These are exactly the commutative noetherian distributive rings. We also give a characterization of c-injective modules over commutative noetherian distributive rings. For a noetherian distributive ring R, we prove that, (1) direct product of simple R-modules is c-injective; (2) an R-module D is c-injective if and only if it is isomorphic to a direct summand of a direct product of simple R-modules and injective R-modules.

• Kyungpook Mathematical Journal
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• v.3 no.2
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• pp.43-48
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• 1960

• Journal of Elementary Mathematics Education in Korea
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• v.18 no.1
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• pp.1-17
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• 2014

Instructions for the commutative property of multiplication at elementary schools tend to be based on checking the equality between the quantities of 'a times b 'and b' times a, ' for example, $3{\times}4=12$ and $4{\times}3=12$. This article critically examined the approaches to teach the commutative property of multiplication from Kant's perspective of mathematical knowledge. According to Kant, mathematical knowledge is a priori. Yet, the numeric exploration by checking the equality between the amounts of 'a groups of b' and 'b groups of a' does not reflect the nature of apriority of mathematical knowledge. I suggest we teach the commutative property of multiplication in a way that it helps reveal the operational schema that is necessarily and generally involved in the transformation from the structure of 'a times b' to the structure of 'b times a.' Distributive reasoning is the mental operation that enables children to perform the structural transformation for the commutative property of multiplication by distributing a unit of one quantity across the other quantity. For example, 3 times 4 is transformed into 4 times 3 by distributing each unit of the quantity 3, which results in $3{\times}4=(1+1+1){\times}4=(1{\times}4)+(1{\times}4)+(1{\times}4)+(1{\times}4)=4+4+4=4{\times}3$. It is argued that the distributive reasoning is also critical in learning the subsequent mathematics concepts, such as (a whole number)${\times}10$ or 100 and fraction concept and fraction multiplication.

• Journal of the Korean Mathematical Society
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• v.34 no.2
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• pp.383-393
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• 1997

In this paper, we study the dependence of corestriction (or transfer) map on the choice of transversals. We also study transfer theorems with respect to some commutative subgroups.