• Honam Mathematical Journal
• /
• v.43 no.2
• /
• pp.199-207
• /
• 2021

In this paper we introduce two classes of modules namely e-symmetric and e-reduced. Also, we study the characterizations of e-symmetric and e-reduced modules and their related properties.

• Journal of the Korean Mathematical Society
• /
• v.45 no.3
• /
• pp.727-740
• /
• 2008

A ring R is called IFP, due to Bell, if ab=0 implies aRb=0 for $a,b{\in}R$. Huh et al. showed that the IFP condition need not be preserved by polynomial ring extensions. But it is shown that ${\sum}^n_{i=0}$ $E_{ai}E$ is a nonzero nilpotent ideal of E whenever R is an IFP ring and $0{\neq}f{\in}F$ is nilpotent, where E is a polynomial ring over R, F is a polynomial ring over E, and $a_i^{'s}$ are the coefficients of f. we shall use the term near IFP to denote such a ring as having place near at the IFPness. In the present note the structures of IFP rings and near-IFP rings are observed, extending the classes of them. IFP rings are NI (i.e., nilpotent elements form an ideal). It is shown that the near-IFPness and the NIness are distinct each other, and the relations among them and related conditions are examined.

• Bulletin of the Korean Mathematical Society
• /
• v.39 no.3
• /
• pp.411-422
• /
• 2002

In this note we are concerned with relationships between one-sided ideals and two-sided ideals, and study the properties of polynomial rings whose maximal one-sided ideals are two-sided, in the viewpoint of the Nullstellensatz on noncommutative rings. Let R be a ring and R[x] be the polynomial ring over R with x the indeterminate. We show that eRe is right quasi-duo for $0{\neq}e^2=e{\in}R$ if R is right quasi-duo; R/J(R) is commutative with J(R) the Jacobson radical of R if R[$\chi$] is right quasi-duo, from which we may characterize polynomial rings whose maximal one-sided ideals are two-sided; if R[x] is right quasi-duo then the Jacobson radical of R[x] is N(R)[x] and so the $K\ddot{o}the's$ conjecture (i.e., the upper nilradical contains every nil left ideal) holds, where N(R) is the set of all nilpotent elements in R. Next we prove that if the polynomial rins R[x], over a reduced ring R with $\mid$X$\mid$ $\geq$ 2, is right quasi-duo, then R is commutative. Several counterexamples are included for the situations that occur naturally in the process of this note.

• Kyungpook Mathematical Journal
• /
• v.51 no.1
• /
• pp.1-10
• /
• 2011

Let (R, T) be a normal pair of commutative rings (i.e., R ${\subseteq}$ T is a unita extension of commutative rings, not necessarily integral domains, such that S is integrally closed in T for each ring S such that R ${\subseteq}$ S ${\subseteq}$ T) such that the total quotient ring of R is a von Neumann regular ring. Let P be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally divided ring. Then R has P if and only if T has P. An example shows that the "if" part of the assertion fails if P is taken to be the "divided domain" property.

• Kyungpook Mathematical Journal
• /
• v.62 no.3
• /
• pp.437-453
• /
• 2022

Unlike for polynomial rings, the notion of multiplication for the near-ring of polynomials is the substitution operation. This leads to somewhat surprising results. Let S be an abelian left near-ring with identity. The relation ~ on S defined by letting a ~ b if and only if ann_S(a) = ann_S(b), is an equivalence relation. The compressed zero-divisor graph 𝚪_E(S) of S is the undirected graph whose vertices are the equivalence classes induced by ~ on S other than [0]_S and [1]_S, in which two distinct vertices [a]_S and [b]_S are adjacent if and only if ab = 0 or ba = 0. In this paper, we are interested in studying the compressed zero-divisor graphs of the zero-symmetric near-ring of polynomials R₀[x] and the near-ring of the power series R₀[[x]] over a commutative ring R. Also, we give a complete characterization of the diameter of these two graphs. It is natural to try to find the relationship between diam(𝚪_E(R₀[x])) and diam(𝚪_E(R₀[[x]])). As a corollary, it is shown that for a reduced ring R, diam(𝚪_E(R)) ≤ diam(𝚪_E(R₀[x])) ≤ diam(𝚪_E(R₀[[x]])).

• Communications of Mathematical Education
• /
• v.10
• /
• pp.283-292
• /
• 2000

We shall introduce new concepts of near-rings, that is, strong reducibility and left semi ${\pi}$-regular near-rings. We will study every strong reducibility of near-ring implies reducibility of near-ring but this converse is not true, and also some characterizations of strong reducibility of near-rings. We shall investigate some relations between strongly reduced near-rings and left strongly regular near-rings, and apply strong reducibility of near-rings to the study of left semi ${\pi}$-regular near-rings, s-weekly regular near-rings and some other regularity of near-rings.

• Kyungpook Mathematical Journal
• /
• v.57 no.3
• /
• pp.401-417
• /
• 2017

Ever since their introduction, skew PBW ($Poincar{\acute{e}}$-Birkhoff-Witt) extensions of rings have kept growing in importance, as researchers characterized their properties (such as primeness, Krull and Goldie dimension, homological properties, etc.) in terms of intrinsic properties of the base ring, and studied their relations with other fields of mathematics, as for example quantum mechanics theory. Many rings and algebras arising in quantum mechanics can be interpreted as skew PBW extensions. Our aim in this paper is to study skew PBW extensions of Baer, quasi-Baer, principally projective and principally quasi-Baer rings, in the case when the base ring R is not assumed to be reduced. We just impose some mild compatibleness over the base ring R, and prove that these properties are stable over this kind of extensions.

• Transactions of the Korean Society of Automotive Engineers
• /
• v.3 no.6
• /
• pp.55-68
• /
• 1995

Approximately 30 to 70 % of the mechanical losses in a reciprocating engine are contributed by the friction at the piston ring-cylinder interface. The friction characteristics of the piston ring during engine operation is known to as mixed lubrication experimentally. The mixed lubrication models based on the Average Reynolds Equation have been used by this time in order to study the tribological performance of the ring. However, the Average Reynolds Equation contains the expected value term(${\bar{h}}_r$) of local film thickness as well as nominal film thickness(h), so that the work of numerically solving ${\bar{h}}_r$ must be included to obtain the pressure in the oil film. The process of solving ${\bar{h}}_T$ causes a greater multiplying in the numerical solution. In this paper the mixed lubrication analysis using the Simplified Average Reynolds Equation in the piston ring is presented. This equation has only h as oil film thickness term. Therefore the tedious numerical procedure required to obtain ${\bar{h}}_T$ is not needed, and also, computation time can be reduced.

• Bulletin of the Korean Mathematical Society
• /
• v.48 no.1
• /
• pp.157-167
• /
• 2011

For a ring endomorphism of a ring R, Krempa called $\alpha$ rigid endomorphism if $a{\alpha}(a)$ = 0 implies a = 0 for a $\in$ R, and Hong et al. called R an $\alpha$-rigid ring if there exists a rigid endomorphism $\alpha$. Due to Rege and Chhawchharia, a ring R is called Armendariz if whenever the product of any two polynomials in R[x] over R is zero, then so is the product of any pair of coefficients from the two polynomials. The Armendariz property of polynomials was extended to one of skew polynomials (i.e., $\alpha$-Armendariz rings and $\alpha$-skew Armendariz rings) by Hong et al. In this paper, we study the relationship between $\alpha$-rigid rings and extended Armendariz rings, and so we get various conditions on the rings which are equivalent to the condition of being an $\alpha$-rigid ring. Several known results relating to extended Armendariz rings can be obtained as corollaries of our results.

• Journal of Korean Society of Water and Wastewater
• /
• v.19 no.1
• /
• pp.98-105
• /
• 2005

The effects of chlorination on the elimination of three estrogenic chemicals such as $17{\beta}$-estradiol (E2), nonylphenol (NP) and bis-phenol A (BPA) were investigated using yeast two-hybrid assay (YTA), estrogen receptor competition assay (ER-CA), and high-performance liquid chromatography/mass spectrometer (LC/MS). Results of YTA, ECA and the analysis of LC/MS indicated that the estrogenic activity of above mentioned three endocrine disruptors were significantly reduced as the result of chlorination. The decrease in estrogenic activity paralleled with decrease in estrogenic chemicals under the influence of free chlorine. One common characteristic of estrogenic chemicals is the presence of a phenolic ring. Considering that a phenolic ring is likely to undergo some sort of transformation in aqueous chlorination solution, the above mentioned results may be applied to the rest of the other estrogenic chemicals in natural waters.