• Bulletin of the Korean Mathematical Society
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• v.36 no.3
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• pp.495-502
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• 1999

The commutative product of the distributions $x^r lnx and $x^{-r-1}$ is evaluated for r=0,1,2,.... The commutative product of the distributions $x^rln(x+i0) and $(x+i0)^{-r-1}$ is also evaluated for r=1,2,.... Further products are deduced.

• Journal of the Chungcheong Mathematical Society
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• v.10 no.1
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• pp.127-139
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• 1997

The spherical non-commutative $\mathbb{S}_{\omega}$ were defined in [2,3]. Assume that no non-trivial matrix algebra can be factored out of the $\mathbb{S}_{\omega}$, and that the fibres are isomorphic to the tensor product of a completely irrational non-commutative torus with a matrix algebra $M_k(\mathbb{C})$. It is shown that the tensor product of the spherical non-commutative torus $\mathbb{S}_{\omega}$ with the even Cuntz algebra $\mathcal{O}_{2d}$ has a trivial bundle structure if and only if k and 2d - 1 are relatively prime, and that the tensor product of the spherical non-commutative torus $S_{\omega}$ with the generalized Cuntz algebra $\mathcal{O}_{\infty}$ has a non-trivial bundle structure when k > 1.

• 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.

• Journal of the Chungcheong Mathematical Society
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• v.11 no.1
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• pp.151-161
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• 1998

The odd spherical non-commutative tori $\mathbb{S}_{\omega}$ were defined in [2]. Assume that no non-trivial matrix algebra can be factored out of $\mathbb{S}_{\omega}$, and that the fibres are isomorphic to the tensor product of a completely irrational non-commutative torus with a matrix algebra $M_{km}(\mathbb{C})$. It is shown that the tensor product of $\mathbb{S}_{\omega}$ with the even Cuntz algebra $\mathcal{O}_{2d}$ has the trivial bundle structure if and, only if km and 2d - 1 are relatively prime, and that the tensor product of $\mathbb{S}_{\omega}$ with the generalized Cuntz algebra $\mathcal{O}_{\infty}$ has a non-trivial bundle structure when km > 1.

• Honam Mathematical Journal
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• v.36 no.1
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• pp.1-9
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• 2014

In this paper, we investigate some properties in commutative diagrams with fiber products of three modules.

• 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.

• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.331-340
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• 1998

We define the spherical non-commutative torus $L_{\omega}$/ as the crossed product obtained by an iteration of l crossed products by actions of, the first action on C( $S^{2n＋l}$). Assume the fibres are isomorphic to the tensor product of a completely irrational non-commutative torus $A_{p}$ with a matrix algebra $M_{m}$ ( ) (m > 1). We prove that $L_{\omega}$/ $M_{p}$ (C) is not isomorphic to C(Prim( $L_{\omega}$/)) $A_{p}$ $M_{mp}$ (C), and that the tensor product of $L_{\omega}$/ with a UHF-algebra $M_{p{\infty}}$ of type $p^{\infty}$ is isomorphic to C(Prim( $L_{\omega}$/)) $A_{p}$ $M_{m}$ (C) $M_{p{\infty}}$ if and only if the set of prime factors of m is a subset of the set of prime factors of p. Furthermore, it is shown that the tensor product of $L_{\omega}$/, with the C＊-algebra K(H) of compact operators on a separable Hilbert space H is not isomorphic to C(Prim( $L_{\omega}$/)) $A_{p}$ $M_{m}$ (C) K(H) if Prim( $L_{\omega}$/) is homeomorphic to $L^{k}$ (n)$\times$ $T^{l'}$ for k and l' non-negative integers (k > 1), where $L^{k}$ (n) is the lens space.$T^{l'}$ for k and l' non-negative integers (k > 1), where $L^{k}$ (n) is the lens space.e.

• Honam Mathematical Journal
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• v.42 no.2
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• pp.219-234
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• 2020

The fundamental relation on a fuzzy hyperring is defined as the smallest equivalence relation, such that the quotient would be the ring, that is not commutative necessarily. In this paper, we introduce a new fuzzy strongly regular equivalence on fuzzy hyperrings, where the ring is commutative with respect to both sum and product. With considering this relation on fuzzy hyperring, the set of the quotient is a commutative ring. Also, we introduce fundamental functor between the category of fuzzy hyperrings and category of commutative rings and some related properties. Eventually, we introduce α-part in fuzzy hyperring and determine some necessary and sufficient conditions so that the relation α is transitive.

• Journal of the Korean Mathematical Society
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• v.43 no.2
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• pp.271-281
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• 2006

The commutative products of the distributions $x^r\;ln^p\;{\mid}x{\mid}\;and\;x^{-r-1}ln^q\;{\mid}x{\mid}$ and of sgn x $x^r\;ln^p\;{\mid}x{\mid}\;and\;sgn\;x\;x^{-r-1}ln^q\;{\mid}x{\mid}$ are evaluated for $r=0,\;\pm1,\;\pm2,...\;and\;p,\;q=0,1,2,....$

• Korean Journal of Mathematics
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• v.14 no.2
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• pp.217-226
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• 2006

In this paper, we investigate the mutual relation among short exact sequences of amalgamated free products which involve augmentation ideals and relation modules. In particular, we find out commutative diagrams having a steady structure in the sense that all of their three columns and rows are short exact sequences.