• Korean Journal of Mathematics
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• v.15 no.2
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• pp.115-119
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• 2007

Let R be a Krull ring with the unique regular maximal ideal M. We show that R has a regular prime element and reg-$dimR=1{\Leftrightarrow}R$ is a factorial ring and reg-$dim(R)=1{\Rightarrow}M$ is invertible ${\Leftrightarrow}R{\varsubsetneq}[R:M]{\Leftrightarrow}M$ is divisorial ${\Leftrightarrow}$ reg-$htM=1{\Rightarrow}R$ is a rank one discrete valuation ring. We also show that if M is generated by regular elements, then R is a rank one discrete valuation ring ${\Rightarrow}$ R is a factorial ring and reg-dim(R)=1.

• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.1-12
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• 2015

Let R be a ring and $D:R^n{\longrightarrow}R$ be n-additive mapping. A map $d:R{\longrightarrow}R$ is said to be the trace of D if $d(x)=D(x,x,{\ldots}x)$ for all $x{\in}R$. Suppose that ${\alpha},{\beta}$ are endomorphisms of R. For any $a,b{\in}R$, let < a, b > $_{({\alpha},{\beta})}=a{\alpha}(b)+{\beta}(b)a$. In the present paper under certain suitable torsion restrictions it is shown that D = 0 if R satisfies either < d(x), $x^m$ > $_{({\alpha},{\beta})}=0$, for all $x{\in}R$ or ${\ll}$ d(x), x > $_{({\alpha},{\beta})}$, $x^m$ > $_{({\alpha},{\beta})}=0$, for all $x{\in}R$. Further, if < d(x), x > ${\in}Z(R)$, the center of R, for all $x{\in}R$ or < d(x)x - xd(x), x >= 0, for all $x{\in}R$, then it is proved that d is commuting on R. Some more related results are also obtained for additive mapping on R.

• Journal of the Korean Mathematical Society
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• v.51 no.2
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• pp.239-250
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• 2014

Let R be a commutative Noetherian (not necessarily local) ring, I an ideal of R and M a finitely generated R-module. In this paper, by computing the local cohomology modules and Ext-modules via the injective resolution of M, we proved that, if for an integer t > 0, dim$_RH_I^i(M){\leq}k$ for ${\forall}i$ < t, then $$\displaystyle\bigcup_{i=0}^{j}(Ass_RH_I^i(M))_{{\geq}k}=\displaystyle\bigcup_{i=0}^{j}(Ass_RExt_R^i(R/I^n,M))_{{\geq}k}$$ for ${\forall}j{\leq}t$ and ${\forall}n$ >0. This shows that${\bigcup}_{n>0}(Ass_RExt_R^i(R/I^n,M))_{{\geq}k}$ is a finite set for ${\forall}i{\leq}t$. Also, we prove that $\displaystyle\bigcup_{i=1}^{r}(Ass_RM/(x_1^{n_1},x_2^{n_2},{\ldots},x_i^{n_i})M)_{{\geq}k}=\displaystyle\bigcup_{i=1}^{r}(Ass_RM/(x_1,x_2,{\ldots},x_i)M)_{{\geq}k}$ if $x_1,x_2,{\ldots},x_r$ is M-sequences in dimension > k and $n_1,n_2,{\ldots},n_r$ are some positive integers. Here, for a subset T of Spec(R), set $T_{{\geq}i}=\{{p{\in}T{\mid}dimR/p{\geq}i}\}$.

• Bulletin of the Korean Mathematical Society
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• v.60 no.2
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• pp.339-348
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• 2023

It is shown that every nilpotent-invariant module can be decomposed into a direct sum of a quasi-injective module and a square-free module that are relatively injective and orthogonal. This paper is also concerned with rings satisfying every cyclic right R-module is nilpotent-invariant. We prove that R ≅ R₁ × R₂, where R₁, R₂ are rings which satisfy R₁ is a semi-simple Artinian ring and R₂ is square-free as a right R₂-module and all idempotents of R₂ is central. The paper concludes with a structure theorem for cyclic nilpotent-invariant right R-modules. Such a module is shown to have isomorphic simple modules eR and fR, where e, f are orthogonal primitive idempotents such that eRf ≠ 0.

• Development and Reproduction
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• v.23 no.1
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• pp.1-9
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• 2019

The ability of oocytes to undergo normal fertilization and embryo development is acquired during oocyte maturation which is transition from the germinal vesicle stage (GV), germinal vesicle breakdown (GVBD) to metaphase of meiosis II (MII). Part of this process includes redistribution of inositol 1, 4, 5-triphosphate receptor (IP3R), a predominant $Ca^{2+}$ channel on the endoplasmic reticulum membrane. Type 1 IP3R (IP3R1) is expressed in mouse oocytes dominantly. At GV stage, IP3R1 are arranged as a network throughout the cytoplasm with minute accumulation around the nucleus. At MII stage, IP3R1 diffuses to the entire cytoplasm in a more reticular manner, and obvious clusters of IP3R1 are observed at the cortex of the egg. This structural reorganization provides acquisition of $[Ca^{2+}]_i$ oscillatory activity during fertilization. In this review, general properties of IP3R1 in somatic cells and mammalian oocyte are introduced.

• 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.

• Kyungpook Mathematical Journal
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• v.46 no.4
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• pp.603-613
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• 2006

Near-rings considered are right near-rings and R is a near-ring. $J_0^r(R)$, the right Jacobson radical of R of type-0, was introduced and studied by the present authors. In this paper $J_1^r(R)$ and $J_2^r(R)$, the right Jacobson radicals of R of type-1 and type-2 are introduced. It is proved that both $J_1^r$ and $J_2^r$ are radicals for near-rings and $J_0^r(R){\subseteq}J_1^r(R){\subseteq}J_2^r(R)$. Unlike the left Jacobson radical classes, the right Jacobson radical class of type-2 contains $M_0(G)$ for many of the finite groups G. Depending on the structure of G, $M_0(G)$ belongs to different right Jacobson radical classes of near-rings. Also unlike left Jacobson-type radicals, the constant part of R is contained in every right 1-modular (2-modular) right ideal of R. For any family of near-rings $R_i$, $i{\in}I$, $J_{\nu}^r({\oplus}_{i{\in}I}R_i)={\oplus}_{i{\in}I}J_{\nu}^r(R_i)$, ${\nu}{\in}\{1,2\}$. Moreover, under certain conditions, for an invariant subnear-ring S of a d.g. near-ring R it is shown that $J_2^r(S)=S{\cap}J_2^r(R)$.

• Bulletin of the Korean Chemical Society
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• v.28 no.11
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• pp.2009-2014
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• 2007

The aminolysis of diphenyl thiophosphinic chloride (2) with substituted anilines in acetonitrile at 55.0 oC is investigated kinetically. Kinetic results yield large Hammett ρX (ρnuc = ?3.97) and Bronsted βX (βnuc = 1.40) values. A concerted mechanism involving a partial frontside nucleophilic attack through a hydrogen-bonded, four-center type transition state is proposed on the basis of the primary normal kinetic isotope effects (kH/kD = 1.0-1.1) with deuterated aniline (XC6H4ND2) nucleophiles. The natural bond order charges on P and the degrees of distortion of 42 compounds: chlorophosphates [(R1O)(R2O)P(=O)Cl], chlorothiophosphates [(R1O)(R2O)P(=S)Cl], phosphonochloridates [(R1O)R2P(=O)Cl], phosphonochlorothioates [(R1O)R2P(=S)Cl], chlorophosphinates [R1R2P(=O)Cl], and chlorothiophosphinates [R1R2P(=S)Cl] are calculated at the B3LYP/ 6-311+G(d,p) level in the gas phase.

• Korean Journal of Mathematics
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• v.21 no.4
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• pp.463-471
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• 2013

Let R be a ring with identity 1, $I(R){\neq}\{0\}$ be the set of all nonunit idempotents in R, and M(R) be the set of all primitive idempotents and 0 of R. We say that I(R) is additive if for all e, $f{\in}I(R)$ ($e{\neq}f$), $e+f{\in}I(R)$. In this paper, the following are shown: (1) I(R) is a finite additive set if and only if $M(R){\backslash}\{0\}$ is a complete set of primitive central idempotents, char(R) = 2 and every nonzero idempotent of R can be expressed as a sum of orthogonal primitive idempotents of R; (2) for a regular ring R such that I(R) is a finite additive set, if the multiplicative group of all units of R is abelian (resp. cyclic), then R is a commutative ring (resp. R is a finite direct product of finite field).

• Proceedings of the Microbiological Society of Korea Conference
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• 2008.05a
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• pp.23-25
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• 2008

Serum complement proteins comprise an important system that is responsible for several innate and adaptive immune defence mechanisms. There were three well described pathways known to lead to the generation of a C3 convertase, which catalyses the proteolysis of complement component C3, and leads to the formation of C3 opsonins (C3b, iC3b and C3d) that fix to bacteria. A pivotal step in the complement pathway is the assembly of a C3 convertase, which digests the C3 complement component to form microbial-binding C3 fragments recognized by leukocytes. The spleen clears microorganisms from the blood. Individuals lacking this organ are more susceptible to Streptococcus pneumoniae. Innate resistance to S. pneumoniae has previously been shown to involve complement components C3 and C4, however this resistance has only a partial requirement for mediators of these three pathways, such as immunoglobulin, factor B and mannose-binding lectin. Therefore it was likely that spleen and complement system provide resistance against blood-borne S. pneumoniae infection through unknown mechanism. To better understand the mechanisms involved, we studied Specific intracellular adhesion molecule-grabbing nonintegrin (SIGN)-R1. SIGN-R1, is a C-type lectin that is expressed at high levels by spleen marginal-zone macrophages and lymph-node macrophages. SIGN-R1 has previously been shown to be the main receptor for bacterial dextrans, as well as for the capsular pneumococcal polysaccharide (CPS) of S. pneumoniae. We examined the specific role of this receptor in the activation of complement. Using a monoclonal antibody that selectively downregulates SIGN-R1 expression in vivo, we show that in response to S. pneumoniae or CPS, SIGN-R1 mediates the immediate proteolysis of C3 and fixation of C3 opsonins to S. pneumoniae or to marginal-zone macrophages that had taken up CPS. These data indicate that SIGN-R1 is largely responsible for the rapid C3 convertase formation induced by S. pneumoniae in the spleen of mice. Also, we found that SIGN-R1 directly binds C1q and that C3 fixation by SIGN-R1 requires C1q and C4 but not factor B or immunoglobulin. Traditionally C3 convertase can be formed by the classical C1q- and immunoglobulin-dependent pathway, the alternative factor-B-dependent pathway and the soluble mannose-binding lectin pathway. Furthermore Conditional SIGN-R1 knockout mice developed deficits in C3 catabolism when given S. pneumoniae or its capsular polysaccharide intravenously. There were marked reductions in proteolysis of serum C3, deposition of C3 on organisms within SIGN-$R1^+$ spleen macrophages, and formation of C3 ligands. The transmembrane lectin SIGN-R1 therefore contributes to innate resistance by an unusual C3 activation pathway. We propose that in the SIGN-R1 mediated complement activation pathway, after binding to polysaccharide, SIGN-R1 captures C1q. SIGN-R1 can then, in association with several other complement proteins including C4, lead to the formation of a C3 convertase and fixation of C3. Therefore, this new pathway for C3 fixation by SIGN-R1, which is unusual as it is a classical C1q-dependent pathway that does not require immuno globulin, contributes to innate immune resistance to certain encapsulated microorganisms.