• East Asian mathematical journal
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• 제16권1호
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• pp.33-48
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• 2000

In this paper, for any ring R with an identity, in order to study prime ideals of the endomorphism ring $End_R$(M) of left R-module $_RM$, meet-prime submodules, prime radical, sum-prime submodules and the prime socle of a module are defined. Some relations of the prime radical, the prime socle of a module and the prime radical of the endomorphism ring of a module are investigated. It is revealed that meet-prime(or sum-prime) modules and semi-meet-prime(or semi-sum-prime) modules have their prime, semi-prime endomorphism rings, respectively.

• Journal of applied mathematics & informatics
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• 제40권1_2호
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• pp.61-68
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• 2022

Let M_p := 2^p - 1 be a Mersenne prime. In this article, we find integers a, b, c, d, e and n satisfying $\sum_{t=0}^{n}\;\({an+b\\ct+d}\)\;=\;M_{p^e}$ given a Mersenne prime number M_p. In order to find a special case that satisfies the above results, we reprove an well-known relation of a certain sum of binomial coefficients and a divisor function.

• 호남수학학술지
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• 제29권4호
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• pp.667-679
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• 2007

We study the prime ideal structure of the direct sum of directed system of rings indexed by a commutative semigroup which is nilpotent extension of a group.

• 대한수학회지
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• 제38권5호
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• pp.987-1030
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• 2001

Some discrimination of modules whose endomorhism rings are prime is introduced, by means of structures of submodules inducing prime ideals of the endomorphism ring End(sub)R (M) of a left R-module (sub)RM over a ring R. Modules with non-prime endomorphism rings are contrapositively studied as well.

• Journal of applied mathematics & informatics
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• 제30권3_4호
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• pp.517-529
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• 2012

In this paper, the existence theorem of two positive solutions is established for nonlinear m-point boundary value problem by using an inequality for the following third-order differential equations $$({\phi}(u^{\prime\prime}))^{\prime}+a(t)f(t,u(t))=0,\;t{\in}(0,1)$$, $${\phi}(u^{\prime\prime}(0))=\sum^{m-2}_{i=1}a_i{\phi}(u^{\prime\prime}({\xi}_i)),\;u^{\prime}(1)=0,\;u(0)=\sum^{m-2}_{i=1}b_iu({\xi}_i)$$, where ${\phi}:R{\rightarrow}R$ is an increasing homeomorphism and homomorphism and $\phi(0)=0$. The nonlinear term f may change sign, as an application, an example to demonstrate our results is given.

• 대한수학회보
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• 제56권4호
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• pp.815-827
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• 2019

It is well known that the denominator of the Dedekind sum s(c, d) divides 2 gcd(d, 3)d and that no smaller denominator independent of c can be expected. In contrast, here we prove that we usually get a smaller denominator in S(H, d), the sum of the s(c, d)'s over all the c's in a subgroup H of order n > 1 in the multiplicative group $(\mathbb{Z}/d\mathbb{Z})^*$. First, we prove that for p > 3 a prime, the sum 2S(H, p) is a rational integer of the same parity as (p-1)/2. We give an application of this result to upper bounds on relative class numbers of imaginary abelian number fields of prime conductor. Finally, we give a general result on the denominator of S(H, d) for non necessarily prime d's. We show that its denominator is a divisor of some explicit divisor of 2d gcd(d, 3).

• 대한수학회보
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• 제58권3호
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• pp.593-602
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• 2021

We use a kind of modified Hurwitz class numbers to express the number of representing a positive integer by some ternary quadratic forms such as the sum of three squares, and the Fourier coefficients of weight 2 modular forms of prime level with trivial character.

• 대한전기학회논문지
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• 제39권12호
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• pp.1288-1295
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• 1990

In this paper, we propose a parallel processing model for the efficient selection of Prime Implicants of Logic Functions. This model consists of simple parallel processing nodes, connections between them, Max Net (a part of Hamming Net) and quasi essential Prime Implicant selection standard in simplified cost form. Through these, this model selects essential Prime Implicants in a certain period of time regardless of the number of given Prime Implicants and minterms and also selects quasi essential Prime Implicants in short time.

• Journal of applied mathematics & informatics
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• 제31권1_2호
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• pp.45-54
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• 2013

Let $$S^{\pm}_{(n,k)}\;:=\{(a,b,x,y){\in}\mathbb{N}^4:ax+by=n,x{\equiv}{\pm}y\;(mod\;k)\}$$. From the formula $\sum_{(a,b,x,y){\in}S^{\pm}_{(n,k)}}\;ab=4\sum_{^{m{\in}\mathbb{N}}_{m<n/k}}\;{\sigma}_1(m){\sigma}_1(n-km)+\frac{1}{6}{\sigma}_3(n)-\frac{1}{6}{\sigma}_1(n)-{\sigma}_3(\frac{n}{k})+n{\sigma}_1(\frac{n}{k})$, we find the Diophantine solutions for modulo $2^{m^{\prime}}$ and $3^{m^{\prime}}$, where $m^{\prime}{\in}\mathbb{N}$.

• 대한수학회논문집
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• 제18권4호
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• pp.615-620
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• 2003

Semi-meet-prime projective modules and fully invariant meet-prime submodules of a projective module are studied. Actually a generalization of the Schur's lemma and semi-prime endmorphism rings of projective modules are considered.