• Kyungpook Mathematical Journal
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• v.55 no.3
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• pp.507-514
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• 2015

Let D be an integral domain, t be the so-called t-operation on D, and S be a (not necessarily saturated) multiplicative subset of D. In this paper, we study the Nagata ring of S-Noetherian domains and locally S-Noetherian domains. We also investigate the t-Nagata ring of t-locally S-Noetherian domains. In fact, we show that if S is an anti-archimedean subset of D, then D is an S-Noetherian domain (respectively, locally S-Noetherian domain) if and only if the Nagata ring $D[X]_N$ is an S-Noetherian domain (respectively, locally S-Noetherian domain). We also prove that if S is an anti-archimedean subset of D, then D is a t-locally S-Noetherian domain if and only if the polynomial ring D[X] is a t-locally S-Noetherian domain, if and only if the t-Nagata ring $D[X]_{N_v}$ is a t-locally S-Noetherian domain.

• Korean Journal of Mathematics
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• v.24 no.3
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• pp.537-543
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• 2016

Let D be an integral domain, X be an indeterminate over D, D[X] be the polynomial ring over D, and D(X) be the Nagata ring of D. Let [d] be the star operation on D[X], which is an extension of the d-operation on D as in [5, Theorem 2.3]. In this paper, we show that D is a sharp domain if and only if D[X] is a [d]-sharp domain, if and only if D(X) is a sharp domain.

• Journal of the Korean Mathematical Society
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• v.58 no.1
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• pp.149-171
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• 2021

Let Cl(A) denote the class group of an arbitrary integral domain A introduced by Bouvier in 1982. Then Cl(A) is the ideal class (resp., divisor class) group of A if A is a Dedekind or a Prüfer (resp., Krull) domain. Let G be an abelian group. In this paper, we show that there is a ring of Krull type D such that Cl(D) = G but D is not a Krull domain. We then use this ring to construct a Prüfer ring of Krull type E such that Cl(E) = G but E is not a Dedekind domain. This is a generalization of Claborn's result that every abelian group is the ideal class group of a Dedekind domain.

• Journal of the Korean Mathematical Society
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• v.53 no.5
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• pp.1057-1075
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• 2016

Let R be a ring in which Nil(R) is a divided prime ideal of R. Then, for a suitable property X of integral domains, we can define a ${\phi}$-X-ring if R/Nil(R) is an X-domain. This device was introduced by Badawi [8] to study rings with zero divisors with a homomorphic image a particular type of domain. We use it to introduce and study a number of concepts such as ${\phi}$-Schreier rings, ${\phi}$-quasi-Schreier rings, ${\phi}$-almost-rings, ${\phi}$-almost-quasi-Schreier rings, ${\phi}$-GCD rings, ${\phi}$-generalized GCD rings and ${\phi}$-almost GCD rings as rings R with Nil(R) a divided prime ideal of R such that R/Nil(R) is a Schreier domain, quasi-Schreier domain, almost domain, almost-quasi-Schreier domain, GCD domain, generalized GCD domain and almost GCD domain, respectively. We study some generalizations of these concepts, in light of generalizations of these concepts in the domain case, as well. Here a domain D is pre-Schreier if for all $x,y,z{\in}D{\backslash}0$, x | yz in D implies that x = rs where r | y and s | z. An integrally closed pre-Schreier domain was initially called a Schreier domain by Cohn in [15] where it was shown that a GCD domain is a Schreier domain.

• Kyungpook Mathematical Journal
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• v.51 no.1
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• pp.1-10
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• 2011

Let (R, T) be a normal pair of commutative rings (i.e., R ${\subseteq}$ T is a unita extension of commutative rings, not necessarily integral domains, such that S is integrally closed in T for each ring S such that R ${\subseteq}$ S ${\subseteq}$ T) such that the total quotient ring of R is a von Neumann regular ring. Let P be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally divided ring. Then R has P if and only if T has P. An example shows that the "if" part of the assertion fails if P is taken to be the "divided domain" property.

• Kyungpook Mathematical Journal
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• v.55 no.4
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• pp.817-826
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• 2015

Let R be a commutative ring with total quotient ring K. Each monomorphic R-module endomorphism of a cyclic R-module is an isomorphism if and only if R has Krull dimension 0. Each monomorphic R-module endomorphism of R is an isomorphism if and only if R = K. We say that R has property (${\star}$) if for each nonzero element $a{\in}R$, each monomorphic R-module endomorphism of R/Ra is an isomorphism. If R has property (${\star}$), then each nonzero principal prime ideal of R is a maximal ideal, but the converse is false, even for integral domains of Krull dimension 2. An integral domain R has property (${\star}$) if and only if R has no R-sequence of length 2; the "if" assertion fails in general for non-domain rings R. Each treed domain has property (${\star}$), but the converse is false.

• Journal of the Optical Society of Korea
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• v.18 no.6
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• pp.766-771
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• 2014

Single-ring all-pass filters with various coupling ratios are designed, fabricated, and characterized to assess the validity of the split-step time-domain modeling approach, which is considered for direct simulation of time-domain characteristics, such as optical delay, of various ring resonator devices. When the coupling ratio of the single-ring all-pass filter is 0.4 and 0.8, the delay time is measured to be about 145 and 42 ps respectively, which is comparable to the time-domain modeling results of 151 and 47 ps respectively. The measurements for two- and three-ring all-pass filters are also found to agree quite well with the simulation results. With these results it is confirmed that the split-step time-domain model could be efficiently incorporated into an optical-communication simulation module for ring resonator delay components in an all-optical packet switching system.

• Journal of applied mathematics & informatics
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• v.39 no.5_6
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• pp.617-622
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• 2021

Let R be a commutative ring with identity, R[X] the polynomial ring over R and S a multiplicative subset of R. Let U = {f ∈ R[X] | f is monic} and let N = {f ∈ R[X] | c(f) = R}. In this paper, we show that if S is an anti-Archimedean subset of R, then R is an S-Noetherian ring if and only if R[X]_U is an S-Noetherian ring, if and only if R[X]_N is an S-Noetherian ring. We also prove that if R is an integral domain and R[X]_U is an S-principal ideal domain, then R is an S-principal ideal domain.

• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.735-744
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• 2024

In this note, we shed new light on Krull domains from the point view of Gorenstein homological algebra. By using the so-called w-operation, we show that an integral domain R is Krull if and only if for any nonzero proper w-ideal I, the Gorenstein global dimension of the w-factor ring (R/I)_w is zero. Further, we obtain that an integral domain R is Dedekind if and only if for any nonzero proper ideal I, the Gorenstein global dimension of the factor ring R/I is zero.

• Asian Pacific Journal of Cancer Prevention
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• v.14 no.2
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• pp.1073-1075
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• 2013

Background: In normal cells, activated epidermal growth factor receptor (EGFR) molecules are subjected to ubiquitination-mediated proteasome degradation pathway by c-Cbl, an ubiquitin ligase that checks uncontrolled proliferation. Hence expression of wild type c-Cbl molecule is essential to keep this degradation machinery in a functional state. Loss of expression or function of c-Cbl may consequently lead to sustained activation of EGFR and promote carcinogenesis, loss of function mutations in the c-Cbl gene already being reported in lung and hematopoietic cancers. However, the genetic status of c-Cbl in oral squamous cell carcinoma (OSCC) is not known. Hence in the present study we investigated the genomic DNA isolated from OSCC tissue biopsy samples for mutations in the RING finger domain coding region of c-Cbl gene, which has also been reported to be most frequently mutated in other cancers. Materials and Methods: Total genomic DNA isolated from thirty two post surgical OSCC tissue samples were amplified using primers flanking the exon 8 of c-Cbl gene that codes for the RING finger domain. The PCR amplicons were then resolved in a 1.2% agarose gel, purified and subjected to direct sequencing to screen for mutations. Results: The sequencing data of the thirty two OSCC samples did not identify mutations in the RING finger domain coding region of c-Cbl gene. Conclusions: To the best of our knowledge, this is the first time that the genetic status of c-Cbl gene in OSCC samples has been investigated. The present data indicates that genetic alteration of RING finger domain coding region of c-Cbl gene is relatively infrequent in OSCC samples.