• Korean Journal of Mathematics
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• v.7 no.2
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• pp.227-233
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• 1999

Let $k$ be a real abelian field and $k_{\infty}={\bigcup}_{n{\geq}0}k_n$ be its $\mathbb{Z}_p$-extension for an odd prime $p$. For each $n{\geq}0$, we denote the class number of $k_n$ by $h_n$. The following is a well known theorem: Theorem. Suppose $p$ remains inert in $k$ and the prime ideal of $k$ above $p$ totally ramifies in $k_{\infty}$. Then $p{\nmid}h_0$ if and only if $p{\nmid}h_n$ for all $n{\geq}0$. The aim of this paper is to generalize above theorem: Theorem 1. Suppose $H^1(G_n,E_n){\simeq}(\mathbb{Z}/p^n\mathbb{Z})^l$, where $l$ is the number of prime ideals of $k$ above $p$. Then $p{\nmid}h_0$ if and only if $p{\nmid}h_n$. Theorem 2. Let $k$ be a real quadratic field. Suppose that $H^1(G_1,E_1){\simeq}(\mathbb{Z}/p\mathbb{Z})^l$. Then $p{\nmid}h_0$ if and only if $p{\nmid}h_n$ for all $n{\geq}0$.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.475-489
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• 2006

After the introduction of fuzzy sets by Zadeh [8], there have been a number of generalizations of fundamental concept. The notion of intuitionistic fuzzy sets introduced by Atanassov is one among them. An intuitionistic fuzzy set is also called a bifuzzy set according to [5]. In this paper, we apply the concept of a bifuzzy set to (implicative) ideals in pseudo MV-algebras. The notion of a bifuzzy (implicative) ideal of a pseudo MV-algebra is introduced, and some related properties are investigated. Conditions for a bifuzzy set to be a bifuzzy ideal are given, and characterizations of a bifuzzy (implicative) ideal are provided. Using a family of ideals, bifuzzy ideals are established.

• Journal of applied mathematics & informatics
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• v.33 no.3_4
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• pp.317-325
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• 2015

Proved that the two topoloies(spectral topology and D−topology) coincide on SpecR, MaxR and MinspecR iff R is complemented, normal and stone ADL respectively.

• 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.56 no.4
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• pp.1077-1097
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• 2019

Let M be a finitely generated module over a regular local ring (R, n). We will fix an n-stable filtration for M and show that the minimal free resolution of M can be obtained from any filtered free resolution of M by zero and negative consecutive cancellations. This result is analogous to [10, Theorem 3.1] in the more general context of filtered free resolutions. Taking advantage of this generality, we will study resolutions obtained by the mapping cone technique and find a sufficient condition for the minimality of such resolutions. Next, we give another application in the graded setting. We show that for a monomial order ${\sigma}$, Betti numbers of I are obtained from those of $LT_{\sigma}(I)$ by so-called zero ${\sigma}$-consecutive cancellations. This provides a stronger version of the well-known cancellation "cancellation principle" between the resolution of a graded ideal and that of its leading term ideal, in terms of filtrations defined by monomial orders.

• Journal of the Korean Society of Safety
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• v.20 no.2 s.70
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• pp.162-168
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• 2005

The lower flash points for the n-propanol+n-decane flammable mixture were measured by the Tag open-cup apparatus(ASTM D 1310). The experimental results of mixture exhibited the lower flash point than those of pure component in the flash point versus composition curve. The experimental value of the minimum flash point is $27^{\circ}C$ at a mole fraction of n-propanol of 0.71, and the flash point of n-propanol was $28^{\circ}C$. The experimentally obtained data were compared with the values that had been calculated by use of the prediction model, which assumes an ideal solution, and the flash point prediction models based on the van Laar equation were used to estimate the activity coefficients. The predictive curve based on an ideal solution deviated from the experimental data for this system. The experimental results demonstrate a close agreement with the predicted curves, which used the van Laar equation. The average absolute deviation(A.A.D.) from using the van Lau equation is $0.83^{\circ}C$. The methodology proposed here in this paper can thus be applied to incorporate an inherently safer design for chemical processes, such as determining safe storage and handling conditions for flammable solutions.

• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1253-1270
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• 2015

Let (R, m) be a commutative Noetherian local domain, M a non-zero finitely generated R-module of dimension n > 0 and I be an ideal of R. In this paper it is shown that if $x_1,{\ldots },x_t$ ($1{\leq}t{\leq}n$) be a sub-set of a system of parameters for M, then the R-module $H^t_{(x_1,{\ldots },x_t)}$(R) is faithful, i.e., Ann $H^t_{(x_1,{\ldots },x_t)}$(R) = 0. Also, it is shown that, if $H^i_I$ (R) = 0 for all i > dim R - dim R/I, then the R-module $H^{dimR-dimR/I}_I(R)$ is faithful. These results provide some partially affirmative answers to the Lynch's conjecture in [10]. Moreover, for an ideal I of an arbitrary Noetherian ring R, we calculate the annihilator of the top local cohomology module $H^1_I(M)$, when $H^i_I(M)=0$ for all integers i > 1. Also, for such ideals we show that the finitely generated R-algebra $D_I(R)$ is a flat R-algebra.

• Journal of the Korean Mathematical Society
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• v.59 no.3
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• pp.571-594
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• 2022

Let D be an integral domain with quotient field K, Pic(D) be the ideal class group of D, and X be an indeterminate. A polynomial overring of D means a subring of K[X] containing D[X]. In this paper, we study almost Dedekind domains which are polynomial overrings of a principal ideal domain D, defined by the intersection of K[X] and rank-one discrete valuation rings with quotient field K(X), and their ideal class groups. Next, let ℤ be the ring of integers, ℚ be the field of rational numbers, and 𝔊f be the set of finitely generated abelian groups (up to isomorphism). As an application, among other things, we show that there exists an overring R of ℤ[X] such that (i) R is a Bezout domain, (ii) R∩ℚ[X] is an almost Dedekind domain, (iii) Pic(R∩ℚ[X]) = $\oplus_{G{\in}G_{f}}$ G, (iv) for each G ∈ 𝔊f, there is a multiplicative subset S of ℤ such that RS ∩ ℚ[X] is a Dedekind domain with Pic(RS ∩ ℚ[X]) = G, and (v) every invertible integral ideal I of R ∩ ℚ[X] can be written uniquely as I = XnQe11···Qekk for some integer n ≥ 0, maximal ideals Qi of R∩ℚ[X], and integers ei ≠ 0. We also completely characterize the almost Dedekind polynomial overrings of ℤ containing Int(ℤ).

• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1253-1259
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• 2011

Let R be a prime ring, I a nonzero ideal of R and n a fixed positive integer. If R admits a generalized derivation F associated with a derivation d such that c for all x, $y{\in}I$. Then either R is commutative or n = 1, d = 0 and F is the identity map on R. Moreover in case R is a semiprime ring and $(F([x,\;y]))^n=[x,\;y]$ for all x, $y{\in}R$, then either R is commutative or n = 1, $d(R){\subseteq}Z(R)$, R contains a non-zero central ideal and for all $x{\in}R$.

• Korean Journal of Mathematics
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• v.24 no.3
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• pp.307-317
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• 2016

Let R be a ring. It is well-known that R is NI if and only if ${\sum}^n_{i=0}Ra_i$ is a nil ideal of R whenever a polynomial ${\sum}^n_{i=0}a_ix^i$ is nilpotent, where x is an indeterminate over R. We consider a condition which is similar to the preceding one: ${\sum}^n_{i=0}Ra_iR$ contains a nonzero nil ideal of R whenever ${\sum}^n_{i=0}a_ix^i$ over R is nilpotent. A ring will be said to be quasi-NI if it satises this condition. The structure of quasi-NI rings is observed, and various examples are given to situations which raised naturally in the process.