• Journal of the Korean Mathematical Society
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• v.42 no.2
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• pp.291-304
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• 2005

We introduce a concept of Cauchy complete Csaszar frames and construct Cauchy completions of Csaszar frames using strict extensions of frames and show that the Cauchy completion gives rise to a coreflection in the categories CsFrm and UCsFrm.

• Communications of the Korean Mathematical Society
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• v.15 no.2
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• pp.371-378
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• 2000

In this paper, we investigate frames that admit a unique uniformity and characterize the completely regular frames which admit a unique uniformity.

• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.539-558
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• 2019

An ideal of a commutative ring is called a d-ideal if it contains the annihilator of the annihilator of each of its elements. Denote by DId(A) the lattice of d-ideals of a ring A. We prove that, as in the case of f-rings, DId(A) is an algebraic frame. Call a ring homomorphism "compatible" if it maps equally annihilated elements in its domain to equally annihilated elements in the codomain. Denote by $SdRng_c$ the category whose objects are rings in which the sum of two d-ideals is a d-ideal, and whose morphisms are compatible ring homomorphisms. We show that $DId:\;SdRng_c{\rightarrow}CohFrm$ is a functor (CohFrm is the category of coherent frames with coherent maps), and we construct a natural transformation $RId{\rightarrow}DId$, in a most natural way, where RId is the functor that sends a ring to its frame of radical ideals. We prove that a ring A is a Baer ring if and only if it belongs to the category $SdRng_c$ and DId(A) is isomorphic to the frame of ideals of the Boolean algebra of idempotents of A. We end by showing that the category $SdRng_c$ has finite products.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.5 no.8
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• pp.1472-1491
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• 2011

As used product transactions are currently on the rise, the demand for transactions of secondhand digital content will grow in the future; thus, learning to make secure transactions while avoiding cyber attacks becomes an important issue. In this paper, we combine the new buyer's secret key, the new buyer's watermark to embed in resold digital content, and the reseller's encrypted watermark, which can prove legal ownership of the reseller. Using the privacy homomorphism property of RSA and exponential calculus, the original seller of digital content can verify the legality of the reseller and the new buyer. We also reduced the load of encryption/decryption digital content using a partial encryption/decryption algorithm to make our protocol more efficient and practical. In the proposed protocol, the seller is not able to conduct piracy and easily frame any other innocent secondhand buyer when a case of piracy is found. In fact, piracy can be clearly traced using the privacy homomorphism property of RSA and the embedded watermark mechanism. Further, in the proposed protocol, the seller himself can trace the piracy using exponential calculus. Since it is unnecessary to trust third party participation, the conspiracy problem is resolved and the new buyer is not required to participate in the dispute. Moreover, the seller, reseller and new buyer can simultaneously benefit from the secondhand transaction.