• Journal of the Korean Mathematical Society
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• v.48 no.2
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• pp.397-420
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• 2011

In this paper we study the structure of closed weakly dense ideals in Privalov spaces $N^p$ (1 < p < $\infty$) of holomorphic functions on the disk $\mathbb{D}$ : |z| < 1. The space $N^p$ with the topology given by Stoll's metric [21] becomes an F-algebra. N. Mochizuki [16] proved that a closed ideal in $N^p$ is a principal ideal generated by an inner function. Consequently, a closed subspace E of $N^p$ is invariant under multiplication by z if and only if it has the form $IN^p$ for some inner function I. We prove that if $\cal{M}$ is a closed ideal in $N^p$ that is dense in the weak topology of $N^p$, then $\cal{M}$ is generated by a singular inner function. On the other hand, if $S_{\mu}$ is a singular inner function whose associated singular measure $\mu$ has the modulus of continuity $O(t^{(p-1)/p})$, then we prove that the ideal $S_{\mu}N^p$ is weakly dense in $N^p$. Consequently, for such singular inner function $S_{\mu}$, the quotient space $N^p/S_{\mu}N^p$ is an F-space with trivial dual, and hence $N^p$ does not have the separation property.

• Bulletin of the Korean Mathematical Society
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• v.30 no.2
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• pp.179-185
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• 1993

In this article S is a topologically complete subspace of $R^{1}$i.e., the relativized topology on S may be metrized so as to make S complete. B(S) is the Borel .sigma.-field of S. For .GAMMA. one takes a set of measurable monotone (increasing or dereasing) functions on S into itself. Make the assumption of pp. There exists $x_{0}$ and a positive integer $n_{0}$ such that (Fig.) It is then shown that there exists a unique inveriant probability to which $p^{(n)}$ (x,dy) converges exponentially fast in a metric (stronger than the Kolmogorov distance); this convergence is uniform for all x .mem. S. This generalizes an earlier result of Bhattacharya and Lee (1988) who considered monotone nondecreasing maps on S.

• Bulletin of the Korean Mathematical Society
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• v.46 no.6
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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).

• Communications of the Korean Mathematical Society
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• v.28 no.1
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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.

• Bulletin of the Korean Mathematical Society
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• v.27 no.2
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• pp.189-196
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• 1990

In this paper, we try to generalize ultraproducts in the category of locally convex spaces. To do so, we introduce D-ultracolimits. It is known [7] that the topology on a non-trivial ultraproduct in the category T $V^{ec}$ of topological vector spaces and continuous linear maps is trivial. To generalize the category Ba $n_{1}$ of Banach spaces and linear contractions, we introduce the category L $C_{1}$ of vector spaces endowed with families of semi-norms closed underfinite joints and linear contractions (see Definition 1.1) and its subcategory, L $C_{2}$ determined by Hausdorff objects of L $C_{1}$. It is shown that L $C_{1}$ contains the category LC of locally convex spaces and continuous linear maps as a coreflective subcategory and that L $C_{2}$ contains the category Nor $m_{1}$ of normed linear spaces and linear contractions as a coreflective subcategory. Thus L $C_{1}$ is a suitable category for the study of locally convex spaces. In L $C_{2}$, we introduce $l_{\infty}$(I. $E_{i}$ ) for a family ( $E_{i}$ )$_{i.mem.I}$ of objects in L $C_{2}$ and then for an ultrafilter u on I. we have a closed subspace $N_{u}$ . Using this, we construct ultraproducts in L $C_{2}$. Using the relationship between Nor $m_{1}$ and L $C_{2}$ and that between Nor $m_{1}$ and Ba $n_{1}$, we show thatour ultraproducts in Nor $m_{1}$ and Ba $n_{1}$ are exactly those in the literatures. For the terminology, we refer to [6] for the category theory and to [8] for ultraproducts in Ba $n_{1}$..