• Communications of the Korean Mathematical Society
• /
• v.19 no.4
• /
• pp.619-637
• /
• 2004

In this paper, a left module over an associative ring with identity is defined to be openly semiprimitive (strongly semiprimitive, respectively) by the zero intersection of all maximal open fully invariant submodules (all maximal open submodules which are fully invariant, respectively) of it. For any projective module, the openly semiprimitivity of the projective module is an equivalent condition of the semiprimitivity of endomorphism ring of the projective module and the strongly semiprimitivity of the projective module is an equivalent condition of the endomorphism ring of the projective module being a sub direct product of a set of subdivisions of division rings.

• Bulletin of the Korean Mathematical Society
• /
• v.49 no.4
• /
• pp.799-813
• /
• 2012

In this paper, we first show that the iteration {$x_n$} defined by $x_{n+1}=P((1-{\alpha}_n)x_n +{\alpha}_nTP[{\beta}_nTx_n+(1-{\beta}_n)x_n])$ converges strongly to some fixed point of T when E is a real uniformly convex Banach space and T is a quasi-nonexpansive non-self mapping satisfying Condition A, which generalizes the result due to Shahzad [11]. Next, we show the strong convergence of the Mann iteration process with errors when E is a real uniformly convex Banach space and T is a quasi-nonexpansive self-mapping satisfying Condition A, which generalizes the result due to Senter-Dotson [10]. Finally, we show that the iteration {$x_n$} defined by $x_{n+1}={\alpha}_nSx_n+{\beta}_nT[{\alpha}^{\prime}_nSx_n+{\beta}^{\prime}_nTx_n+{\gamma}^{\prime}_n{\upsilon}_n]+{\gamma}_nu_n$ converges strongly to a common fixed point of T and S when E is a real uniformly convex Banach space and T, S are two quasi-nonexpansive self-mappings satisfying Condition D, which generalizes the result due to Ghosh-Debnath [3].

• Honam Mathematical Journal
• /
• v.29 no.4
• /
• pp.555-567
• /
• 2007

We first find strongly 2-primal rings whose sub direct product is not (strongly) 2-primal. Moreover we observe some kinds of ring extensions of (strongly) 2-primal rings. As an example we show that if R is a ring and M is a multiplicative monoid in R consisting of central regular elements, then R is strongly 2-primal if and only if so is $RM^{-1}$. Various properties of (strongly) 2-primal rings are also studied.

• Communications of the Korean Mathematical Society
• /
• v.39 no.1
• /
• pp.93-104
• /
• 2024

Let R be a commutative ring with identity. In this paper, we introduce a new class of ideals called the class of strongly quasi J-ideals lying properly between the class of J-ideals and the class of quasi J-ideals. A proper ideal I of R is called a strongly quasi J-ideal if, whenever a, b ∈ R and ab ∈ I, then a² ∈ I or b ∈ Jac(R). Firstly, we investigate some basic properties of strongly quasi J-ideals. Hence, we give the necessary and sufficient conditions for a ring R to contain a strongly quasi J-ideals. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the primary ideals, the prime ideals and the maximal ideals. Finally, we give an idea about some strongly quasi J-ideals of the quotient rings, the localization of rings, the polynomial rings and the trivial rings extensions.

• Honam Mathematical Journal
• /
• v.23 no.1
• /
• pp.21-28
• /
• 2001

In this short paper, we generalize some theorems about pseudo valuation domain to ring and give characterizations of psedo valuation ring.

• Korean Journal of Materials Research
• /
• v.28 no.10
• /
• pp.583-589
• /
• 2018

This study investigates the effect of fatigue stress on the damping capacity in a damaged Fe-22Mn-12Cr-3Ni-2Si-4Co damping alloy under fatigue stress. ${\alpha}^{\prime}$ and ${\varepsilon}-martensite$ forms by fatigue stress in the damaged Fe-22Mn-12Cr-3Ni-2Si4-Co damping alloy under fatigue stress. The ${\alpha}^{\prime}$ and ${\varepsilon}-martensite$ forms with the specific direction and surface relief, or they cross each other. With an increasing fatigue stress, the volume fraction of ${\alpha}^{\prime}-martensite$ and ${\varepsilon}-martensite$ increases. With an increasing fatigue stress, the damping capacity increases with an increase in the volume fraction of ${\varepsilon}-martensite$. The increase in the damping capacity in the damaged Fe-22Mn-12Cr-3Ni-2Si-4Co alloy under fatigue stress strongly affects the increase of ${\varepsilon}-martensite$ formed by fatigue stress, but the damping capacity of the damaged Fe-22Mn-12Cr-3Ni-2Si-4Co damping alloy under fatigue stress is strongly controlled by a large amount of ${\alpha}^{\prime}-martensite$.

• Communications of the Korean Mathematical Society
• /
• v.22 no.3
• /
• pp.323-330
• /
• 2007

In this paper we introduce the notion of strongly regular near-subtraction semigroups (right). We have shown that a near-subtraction semigroup X is strongly regular if and only if it is regular and without non zero nilpotent elements. We have also shown that in a strongly regular near-subtraction semigroup X, the following holds: (i) Xa is an ideal for every a $\in$ X (ii) If P is a prime ideal of X, then there exists no proper k-ideal M such that P $\subset$ M (iii) Every ideal I of X fulfills $I=I^2$.

• Bulletin of the Korean Mathematical Society
• /
• v.55 no.3
• /
• pp.913-920
• /
• 2018

Let R be a 2-primal strongly 2-nil-clean ring. We prove that every square matrix over R is the sum of a tripotent and a nilpotent matrices. The similar result for rings of bounded index is proved. We thereby provide a large class of rings over which every matrix is the sum of a tripotent and a nilpotent matrices.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.2
• /
• pp.505-514
• /
• 2019

Let R be a commutative ring with identity and M be a unitary R-module. A submodule N of M is called a dense submodule if $Hom_R(M/N,\;E_R(M))=0$, where $E_R(M)$ is the injective hull of M. The R-module M is said to be monoform if any nonzero submodule of M is a dense submodule. In this paper, among the other results, it is shown that any kind of the following module is monoform. (1) The prime R-module M such that for any nonzero submodule N of M, $Ann_R(M/N){\neq}Ann_R(M)$. (2) Strongly prime R-module. (3) Faithful multiplication module over an integral domain.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.3
• /
• pp.729-743
• /
• 2019

In this paper, we introduce strongly quasi primary ideals which is an intermediate class of primary ideals and quasi primary ideals. Let R be a commutative ring with nonzero identity and Q a proper ideal of R. Then Q is called strongly quasi primary if $ab{\in}Q$ for $a,b{\in}R$ implies either $a^2{\in}Q$ or $b^n{\in}Q$ ($a^n{\in}Q$ or $b^2{\in}Q$) for some $n{\in}{\mathbb{N}}$. We give many properties of strongly quasi primary ideals and investigate the relations between strongly quasi primary ideals and other classical ideals such as primary, 2-prime and quasi primary ideals. Among other results, we give a characterization of divided rings in terms of strongly quasi primary ideals. Also, we construct a subgraph of ideal based zero divisor graph ${\Gamma}_I(R)$ and denote it by ${\Gamma}^*_I(R)$, where I is an ideal of R. We investigate the relations between ${\Gamma}^*_I(R)$ and ${\Gamma}_I(R)$. Further, we use strongly quasi primary ideals and ${\Gamma}^*_I(R)$ to characterize von Neumann regular rings.