• 대한수학회논문집
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• 제38권2호
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• pp.451-459
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• 2023

We give a characterization of zero divisors of the ring C[a, b]. Using the Weierstrass approximation theorem, we completely characterize topological divisors of zero of the Banach algebra C[a, b]. We also characterize the zero divisors and topological divisors of zero in ℓ^∞. Further, we show that zero is the only zero divisor in the disk algebra 𝒜 (𝔻) and that the class of singular elements in 𝒜 (𝔻) properly contains the class of topological divisors of zero. Lastly, we construct a class of topological divisors of zero of 𝒜 (𝔻) which are not zero divisors.

• Journal of applied mathematics & informatics
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• 제42권3호
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• pp.663-680
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• 2024

The zero divisor graph is the most basic way of representing an algebraic structure as a graph. For any commutative ring R, each element is a vertex on the zero divisor graph and two vertices are defined as adjacent if and only if the product of those vertices equals zero. In this research, we determine some topological indices such as the Wiener index, the edge-Wiener index, the hyper-Wiener index, the Harary index, the first Zagreb index, the second Zagreb index, and the Gutman index of zero divisor graph of integers modulo prime power and its direct product.

• 대한수학회보
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• 제43권3호
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• pp.653-655
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• 2006

We give a simple proof of the statement that if T is a bounded linear operator on a complex HIlbert space then T is Fredholm if and only if ${\pi}(T)$ is not a TDZ, where ${\pi}(\cdot)$ is the Catkin homomorphism.

• 대한수학회논문집
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• 제28권1호
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• pp.79-85
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• 2013

The structure of a poset P with smallest element 0 is looked at from two view points. Firstly, with respect to the Zariski topology, it is shown that Spec(P), the set of all prime semi-ideals of P, is a compact space and Max(P), the set of all maximal semi-ideals of P, is a compact $T_1$ subspace. Various other topological properties are derived. Secondly, we study the semi-ideal-based zero-divisor graph structure of poset P, denoted by $G_I$ (P), and characterize its diameter.

• East Asian mathematical journal
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• 제39권3호
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• pp.349-354
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• 2023

The motivation in this paper comes from the recent results about Bell inequalities and topological insulators from group theory. Symmetries which are interested in group theory could be mainly used to find material structures. In this point of views, we study group extending by adding one relator which is easily called an equation. So a relative group extension by a adding relator is aspherical if the natural injection is one-to-one and the group ring has no zero divisor. One of concepts of asphericity means that a new group by a adding relator is well extended. Also, we consider that several equations and relative presentations over torsion-free groups are related to zero divisors.

• 대한수학회보
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• 제46권6호
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• pp.1051-1060
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• 2009

In this paper, we give topological properties of collection of prime ideals in 2-primal near-rings. We show that Spec(N), the spectrum of prime ideals, is a compact space, and Max(N), the maximal ideals of N, forms a compact $T_1$-subspace. We also study the zero-divisor graph $\Gamma_I$(R) with respect to the completely semiprime ideal I of N. We show that ${\Gamma}_{\mathbb{P}}$ (R), where $\mathbb{P}$ is a prime radical of N, is a connected graph with diameter less than or equal to 3. We characterize all cycles in the graph ${\Gamma}_{\mathbb{P}}$ (R).