• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.117-126
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• 2020

Let R be a commutative ring with identity. A proper ideal I of R is called 2-prime if for all a, b ∈ R such that ab ∈ I, then either a² or b² lies in I. In this paper, we study 2-prime ideals which are generalization of prime ideals. Our study provides an analogous to the prime avoidance theorem and some applications of this theorem. Also, it is shown that if R is a PID, then the families of primary ideals and 2-prime ideals of R are identical. Moreover, a number of examples concerning 2-prime ideals are given. Finally, rings in which every 2-prime ideal is a prime ideal are investigated.

• Proceedings of the Korean Ceranic Society Conference
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• 2000.04a
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• pp.117.2-117.2
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• 2000

• Journal of applied mathematics & informatics
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• v.40 no.1_2
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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.

• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.897-908
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• 2021

In this paper, we introduce the notion of pseudo 2-prime ideals in commutative rings. Let R be a commutative ring with a nonzero identity. A proper ideal P of R is said to be a pseudo 2-prime ideal if whenever xy ∈ P for some x, y ∈ R, then x²ⁿ ∈ Pⁿ or y²ⁿ ∈ Pⁿ for some n ∈ ℕ. Various examples and properties of pseudo 2-prime ideals are given. We also characterize pseudo 2-prime ideals of PID's and von Neumann regular rings. Finally, we use pseudo 2-prime ideals to characterize almost valuation domains (AV-domains).

• The Pure and Applied Mathematics
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• v.30 no.3
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• pp.337-345
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• 2023

Different versions of the boundary Schwarz lemma for the 𝒩 (𝜌) class are discussed in this study. Also, for the function g(z) = z+b₂z²+b₃z³+... defined in the unit disc D such that g ∈ 𝒩 (𝜌), we estimate a modulus of the angular derivative of g(z) function at the boundary point 1 ∈ 𝜕D with g'(1) = 1 + 𝜎 (1 - 𝜌), where ${\rho}={\frac{1}{n}}{\sum\limits_{i=1}^{n}}g(c_i)={\frac{g^{\prime}(c_1)+g^{\prime}(c_2)+{\ldots}+g^{\prime}(c_n)}{n}}{\in}g^{\prime}(D)$ and 𝜌≠1, 𝜎 > 1 and c₁, c₂, ..., c_n ∈ 𝜕D. That is, we shall give an estimate below |g"(1)| according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and z ≠ 0. Estimating is made by using the arithmetic average of n different derivatives g'(c₁), g'(c₂), ..., g'(c_n).

• Bulletin of the Korean Chemical Society
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• v.22 no.9
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• pp.1033-1036
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• 2001

• Kyungpook Mathematical Journal
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• v.56 no.4
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• pp.1085-1101
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• 2016

In this paper, all rings are commutative with nonzero identity. Let M be an R-module. A proper submodule N of M is called a classical prime submodule, if for each $m{\in}M$ and elements a, $b{\in}R$, $abm{\in}N$ implies that $am{\in}N$ or $bm{\in}N$. We introduce the concept of "weakly classical prime submodules" and we will show that this class of submodules enjoys many properties of weakly 2-absorbing ideals of commutative rings. A proper submodule N of M is a weakly classical prime submodule if whenever $a,b{\in}R$ and $m{\in}M$ with $0{\neq}abm{\in}N$, then $am{\in}N$ or $bm{\in}N$.

• Bulletin of the Korean Mathematical Society
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• v.55 no.2
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• pp.573-586
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• 2018

Let R be a noncommutative prime ring of characteristic different from 2, Q be its maximal right ring of quotients and C be its extended centroid. Suppose that $f(x_1,{\ldots},x_n)$ be a noncentral multilinear polynomial over $C,b{\in}Q,F$ a b-generalized derivation of R and d is a nonzero derivation of R such that d([F(f(r)), f(r)]) = 0 for all $r=(r_1,{\ldots},r_n){\in}R^n$. Then one of the following holds: (1) there exists ${\lambda}{\in}C$ such that $F(x)={\lambda}x$ for all $x{\in}R$; (2) there exist ${\lambda}{\in}C$ and $p{\in}Q$ such that $F(x)={\lambda}x+px+xp$ for all $x{\in}R$ with $f(x_1,{\ldots},x_n)^2$ is central valued in R.

• IMMUNE NETWORK
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• v.5 no.2
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• pp.89-98
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• 2005

Background: DNA vaccination represents an anticipated approach for the control of numerous infectious diseases. Used alone, however, DNA vaccine is weak immunogen inferior to viral vectors. In recent, heterologous prime-boost vaccination leads DNA vaccines to practical reality. Methods: We assessed prime-boost immunization strategies with a DNA vaccine (minigene, $gB_{498-505}$ DNA) and recombinant vaccinia virus $(vvgB_{498-505})$ expressing epitope $gB_{498-505}$ (SSIEF ARL) of CD8+ T cells specific for glycoprotein B (gB) of herpes simplex virus (HSV). Animals were immunized primarily with $gB_{498-505}$ epitope-expressing DNA vaccine/recombinant vaccinia virus and boosted with alternative vaccine type expressing entire Ag. Results: In prime-boost protocols using vvgBw (recombinant vaccinia virus expressing entire Ag) and $vvgB_{498-505}$, CD8+ T cell-mediated immunity was induced maximally at both acute and memory stages if primed with vvgBw and boosted with $vvgB_{498-505}$ as evaluated by CTL activity, intracellular IFN-staining, and MHC class I tetramer staining. Similarly $gB_{498-505}$ DNA prime-gBw DNA (DNA vaccine expressing entire Ag) boost immunization elicited the strongest CD8+ T cell responses in protocols based on DNA vaccine. However, the level of CD8+ T cell-mediated immunity induced with prime-boost vaccination using DNA vaccine expressing epitope or entire Ag was inferior to those based on vvgBw and $vvgB_{498-505}$. Of particular interest CD8+ T cell-mediated immunity was optimally induced when $vvgB_{498-505}$ was used to prime and gB DNA was used as alternative boost. Especially CD7+ T cell responses induced by such protocol was longer lasted than other protocols. Conclusion: These facts direct to search for the effective strategy to induce optimal CD8+ T cell-mediated immunity against cancer and viral infection.

• Bulletin of the Korean Mathematical Society
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• v.30 no.2
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• pp.187-191
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• 1993

In this note we prove the following result: Let A be a complex Banach *-algebra with continuous involution and let B be an $A^{*}$-algebra./T(A) = B. Then T is continuous (Theorem 2). From above theorem some others results of special interest and some well-known results follow. (Corollaries 3,4,5,6 and 7). We close this note with some generalizations and some remarks (Theorems 8.9.10 and question). Throughout this note we consider only complex algebras. Let A and B be complex algebras. A linear mapping T from A into B is called jordan homomorphism if T( $x^{1}$) = (Tx)$^{2}$ for all x in A. A linear mapping T : A .rarw. B is called spectrally-contractive mapping if .rho.(Tx).leq..rho.(x) for all x in A, where .rho.(x) denotes spectral radius of element x. Any homomorphism algebra is a spectrally-contractive mapping. If A and B are *-algebras, then a homomorphism T : A.rarw.B is called *-homomorphism if (Th)$^{*}$=Th for all self-adjoint element h in A. Recall that a Banach *-algebras is a complex Banach algebra with an involution *. An $A^{*}$-algebra A is a Banach *-algebra having anauxiliary norm vertical bar . vertical bar which satisfies $B^{*}$-condition vertical bar $x^{*}$x vertical bar = vertical bar x vertical ba $r^{2}$(x in A). A Banach *-algebra whose norm is an algebra $B^{*}$-norm is called $B^{*}$-algebra. The *-semi-simple Banach *-algebras and the semi-simple hermitian Banach *-algebras are $A^{*}$-algebras. Also, $A^{*}$-algebras include $B^{*}$-algebras ( $C^{*}$-algebras). Recall that a semi-prime algebra is an algebra without nilpotents two-sided ideals non-zero. The class of semi-prime algebras includes the class of semi-prime algebras and the class of prime algebras. For all concepts and basic facts about Banach algebras we refer to [2] and [8].].er to [2] and [8].].