• Bulletin of the Korean Mathematical Society
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• v.57 no.2
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• pp.419-427
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• 2020

In this paper, the projectivity of finitely generated flat modules of a commutative ring are studied from a topological point of view. Then various interesting results are obtained. For instance, it is shown that if a ring has either finitely many minimal primes or finitely many maximal ideals then every finitely generated flat module over it is projective. It is also shown that if a particular subset of the prime spectrum of a ring satisfies some certain ascending or descending chain conditions, then every finitely generated flat module over this ring is projective. These results generalize some major results in the literature on the projectivity of finitely generated flat modules.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.3
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• pp.467-480
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• 2009

In 1859, Bernhard Riemann, in his epoch-making memoir, extended the Euler zeta function $\zeta$(s) (s > 1; $s{\in}\mathbb{R}$) to the Riemann zeta function $\zeta$(s) ($\Re$(s) > 1; $s{\in}\mathbb{C}$) to investigate the pattern of the primes. Sine the time of Euler and then Riemann, the Riemann zeta function $\zeta$(s) has involved and appeared in a variety of mathematical research subjects as well as the function itself has been being broadly and deeply researched. Among those things, we choose to make a further investigation of the following subjects: Evaluation of $\zeta$(2k) ($k {\in}\mathbb{N}$); Approximate functional equations for $\zeta$(s); Series involving the Riemann zeta function.

• Bulletin of the Korean Mathematical Society
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• v.52 no.6
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• pp.1911-1924
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• 2015

A cyclotomic polynomial ${\Phi}_n(x)$ is said to be ternary if n = pqr for three distinct odd primes p < q < r. Let A(n) be the largest absolute value of the coefficients of ${\Phi}_n(x)$. If A(n) = 1 we say that ${\Phi}_n(x)$ is flat. In this paper, we classify all flat ternary cyclotomic polynomials ${\Phi}_{pqr}(x)$ in the case $q{\equiv}{\pm}1$ (mod p) and $4r{\equiv}{\pm}1$ (mod pq).

• Korean Journal of Mathematics
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• v.5 no.1
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• pp.61-74
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• 1997

Let $d$ be a positive integer with $d\not{\equiv}2$ mod 4, and let $K=\mathbb{Q}({\zeta}_{pd})$ for S an odd prime $p$ such that $p{\equiv}1$ mod $d$. Let $K_{\infty}={\bigcup}_{n{\geq}0}K_n$ be the cyclotomic $\mathbb{Z}_p$-extension of $K=K_0$. In this paper, explicit generators for the Tate cohomology group $\hat{H}^{-1}$($G_{m,n}$ are given when $d=qr$ is a product of two distinct primes, where $G_{m,n}$ is the Galois group Gal($K_m/K_n$) and $C_m$ is the group of cyclotomic units of $K_m$. This generalizes earlier results when $d=q$ is a prime.

• Korean Journal of Mathematics
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• v.22 no.4
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• pp.725-742
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• 2014

In this paper, we find all inequivalent classes of self-orthogonal codes over $Z_{p^2}$ of lengths $l{\leq}3$ for all primes p, using similar method as in [3]. We find that the classification of self-orthogonal codes over $Z_{p^2}$ includes the classification of all codes over $Z_p$. Consequently, we classify all the codes over $Z_p$ and self-orthogonal codes over $Z_{p^2}$ of lengths $l{\leq}3$ according to the automorphism group of each code.

• Journal of Integrative Natural Science
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• v.4 no.4
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• pp.278-281
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• 2011

We discuss an application of 'restrictions on the entries of the maps in the minimal free resolution' and '$SC_r$-condition of modules', and give an alternative proof of the following result of Foxby: Let M be a finitely generated module of dimension over a Noetherian local ring (A,m). Suppose that $\hat{A}$ has no embedded primes. If A is not Gorenstein, then ${\mu}_i(m,A){\geq}2$ for all i ${\geq}$ dimA.

• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.737-743
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• 2016

Let a and b be two ideals of a commutative Noetherian ring R, M a finitely generated R-module and i an integer. In this paper we study formal local cohomology modules with respect to a pair of ideals. We denote the i-th a-formal local cohomology module M with respect to b by ${\mathfrak{F}}^i_{a,b}(M)$. We show that if ${\mathfrak{F}}^i_{a,b}(M)$ is artinian, then $a{\subseteq}{\sqrt{(0:{\mathfrak{F}}^i_{a,b}(M))$. Also, we show that ${\mathfrak{F}}^{\text{dim }M}_{a,b}(M)$ is artinian and we determine the set $Att_R\;{\mathfrak{F}}^{\text{dim }M}_{a,b}(M)$.

• East Asian mathematical journal
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• v.29 no.1
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• pp.39-52
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• 2013

Let (X, S) be an association scheme where X is a finite set and S is a partition of $X{\times}X$. The size of X is called the order of (X, S). We define $\mathcal{C}$ to be the set of positive integers m such that each association scheme of order $m$ is commutative. It is known that each prime is belonged to $\mathcal{C}$ and it is conjectured that each prime square is belonged to $\mathcal{C}$. In this article we give a sufficient condition for a scheme of order pq to be commutative where $p$ and $q$ are primes, and obtain a partial answer for the conjecture in case where $p=q$.

• Journal of Microbiology
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• v.33 no.3
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• pp.171-177
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• 1995

Random amplified polymorphic DAN markers were characterized for three taxonomically problematic Penicillium species : P. aurantiogriseum var. Aurantiogriseum, P. verrucosum and P. puberulum, as well as for 25 species of mono, bi-, and terverticillate Penicillia. The relationships among mono, bi-, and terverticillate Penicillium species were determined from these RAPD markers. Eight species from mono-, eight from bi-, and nine from terverticilate Penicillia were examined. With 14 randomly chosen 10-mer primes, a 310 character by 25 species matrix was generated. Phenetic analysis separated the 25 species into three genetically distinct groups that correspond to the different arrangements of penicilli (mono-, bi-, and terverticillate). The results of this study suggest that P. aurantiogriseum var. aurantiogriseum, P. VERRUCOSUM, AND P. puberulum represent genetically distinct species, and that P. vulpinum should be included in terverticilate Penicillia. Phenogram branching patterns indicated that biverticillate species are genetically more similar to monoverticilate species than they are to terverticillate species.

• The Pure and Applied Mathematics
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• v.8 no.2
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• pp.145-152
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• 2001

Posner [Proc. Amer. Math. Soc. 8 (1957), 1093-1100] defined a derivation on prime rings and Herstein [Canad, Math. Bull. 21 (1978), 369-370] derived commutative property of prime ring with derivations. Recently, Bergen [Canad. Math. Bull. 26 (1983), 267-227], Bell and Daif [Acta. Math. Hunger. 66 (1995), 337-343] studied derivations in primes and semiprime rings. Also, in near-ring theory, Bell and Mason [Near-Rungs and Near-Fields (pp. 31-35), Proceedings of the conference held at the University of Tubingen, 1985. Noth-Holland, Amsterdam, 1987; Math. J. Okayama Univ. 34 (1992), 135-144] and Cho [Pusan Kyongnam Math. J. 12 (1996), no. 1, 63-69] researched derivations in prime and semiprime near-rings. In this paper, Posner, Bell and Mason＇s results are extended in prime near-rings with some conditions.