• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.1913-1925
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• 2017

Let R be a ring with identity. An ideal N of R is called ideal-symmetric (resp., ideal-reversible) if $ABC{\subseteq}N$ implies $ACB{\subseteq}N$ (resp., $AB{\subseteq}N$ implies $BA{\subseteq}N$) for any ideals A, B, C in R. A ring R is called ideal-symmetric if zero ideal of R is ideal-symmetric. Let S(R) (called the ideal-symmetric radical of R) be the intersection of all ideal-symmetric ideals of R. In this paper, the following are investigated: (1) Some equivalent conditions on an ideal-symmetric ideal of a ring are obtained; (2) Ideal-symmetric property is Morita invariant; (3) For any ring R, we have $S(M_n(R))=M_n(S(R))$ where $M_n(R)$ is the ring of all n by n matrices over R; (4) For a quasi-Baer ring R, R is semiprime if and only if R is ideal-symmetric if and only if R is ideal-reversible.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.22 no.3
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• pp.557-561
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• 1997

In this paper, we present three new classes of binary pseudorandom sequences of period ${2_n}-1$ with ideal autocorrelation. These sequences are newly found by an extensive computer search, and the conjectures on the construction of these sequences are formulated. We also classify the binary sequences of period ${2_n}-1$ with ideal autocorrelation, and enumerate them.

• East Asian mathematical journal
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• v.18 no.2
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• pp.271-282
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• 2002

Let R be a ring with an endomorphism $\sigma$ and a derivation $\delta$. An ideal I of R is ($\sigma,\;\delta$)-ideal of R if $\sigma(I){\subseteq}I$ and $\delta(I){\subseteq}I$. An ideal P of R is a ($\sigma,\;\delta$)-prime ideal of R if P(${\neq}R$) is a ($\sigma,\;\delta$)-ideal and for ($\sigma,\;\delta$)-ideals I and J of R, $IJ{\subseteq}P$ implies that $I{\subseteq}P$ or $J{\subseteq}P$. An ideal Q of R is ($\sigma,\;\delta$)-semiprime ideal of R if Q is a ($\sigma,\;\delta$)-ideal and for ($\sigma,\;\delta$)-ideal I of R, $I^2{\subseteq}Q$ implies that $I{\subseteq}Q$. The ($\sigma,\;\delta$)-prime radical (resp. prime radical) is defined by the intersection of all ($\sigma,\;\delta$)-prime ideals (resp. prime ideals) of R and is denoted by $P_{(\sigma,\delta)}(R)$(resp. P(R)). In this paper, the following results are obtained: (1) $P_{(\sigma,\delta)}(R)$ is the smallest ($\sigma,\;\delta$)-semiprime ideal of R; (2) For every extended endomorphism $\bar{\sigma}$ of $\sigma$, the $\bar{\sigma}$-prime radical of an Ore extension $P(R[x;\sigma,\delta])$ is equal to $P_{\sigma,\delta}(R)[x;\sigma,\delta]$.

• East Asian mathematical journal
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• v.24 no.4
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• pp.377-387
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• 2008

We consider associative ring R (not necessarily commutative). In this paper the concepts: zero square ring of type-1/type-2, zero square ideal of type-1/type-2, zero square dimension of a ring R were introduced and obtained several important results. Finally, some relations between the zero square dimension of the direct sum of finite number of rings; and the sum of the zero square dimension of individual rings; were obtained. Necessary examples were provided.

• Journal of applied mathematics & informatics
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• v.10 no.1_2
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• pp.251-259
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• 2002

The fuzzification of an ideal in a Lie algebra is considered. Using a level subset of a fuzzy subset of a Lie algebra, we give a characterization of a fuzzy ideal, and using a family of ideals of a Lie algebra, we establish a fuzzy ideal. With relation to the ascending chain of ideals, a characterization for the set of values of any fuzzy ideal to be a well-ordered subset of the closed unit interval [0,1] is stated.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.971-983
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• 2016

A partially ordered set P is ideal-homogeneous provided that for any ideals I and J, if $$I{\sim_=}_{\sigma}J$$, then there exists an automorphism ${\sigma}^*$ such that ${\sigma}^*{\mid}_I={\sigma}$. Behrendt [1] characterizes the ideal-homogeneous partially ordered sets of height 1. In this paper, we characterize the ideal-homogeneous partially ordered sets of height 2 and nd some families of ideal-homogeneous partially ordered sets.

• Kyungpook Mathematical Journal
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• v.52 no.1
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• pp.91-97
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• 2012

Let $R$ be a commutative semiring. We define a proper ideal $I$ of $R$ to be 2-absorbing (resp., weakly 2-absorbing) if $abc{\in}I$ (resp., $0{\neq}abc{\in}I$) implies $ab{\in}I$ or $ac{\in}I$ or $bc{\in}I$. We show that a weakly 2-absorbing ideal $I$ with $I^3{\neq}0$ is 2-absorbing. We give a number of results concerning 2-absorbing and weakly 2-absorbing ideals and examples of weakly 2-absorbing ideals. Finally we de ne the concept of 0 - (1-, 2-, 3-)2-absorbing ideals of $R$ and study the relationship among these classes of ideals of $R$.

• International Journal of Human Ecology
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• v.7 no.2
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• pp.85-95
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• 2006

The purpose of this study was to examine mothers' ideal and actual parenting behaviors toward their infants in three parenting domains; social, didactic, and limit setting. A total of 264 mothers of young children under age three from Seoul, Korea completed Parental Style Questionnaires (PSQ). Mothers' self report on their ideal and actual parenting were explored as a function of child sex, age, and birth order. As expected, there were significant differences between mothers' ideal and actual behaviors in all three parenting domains: Mothers' ideal behaviors such as social interaction, didactic interaction and limit setting were higher than those of their actual behaviors. For mothers' ideal parenting, results revealed neither significant main effects nor interaction effects. However, the Parenting Domain x Birth-Order 2-way interaction and the Parenting Domain x Child Age 2-way interaction were significant for mothers' actual behaviors. Specifically, mothers reported more social and didactic behaviors with their first-born than later born children, but not for limit setting behavior. It was also found that higher limit setting behaviors were apparent for their 2- and 3-year-old than 1-year old children whereas lower social interactions were found for 3-year-old than for 1-year-old. In light of universality and uniqueness, mothers' parenting behavior toward young children has been discussed.

• Journal of applied mathematics & informatics
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• v.34 no.5_6
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• pp.405-419
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• 2016

The notion of Γ-semiring was introduced by M. Murali Krishna Rao [8] as a generalization of Γ-ring as well as of semiring. In this paper fuzzy k-ideals, k-fuzzy ideals and fuzzy-2-prime ideals in Γ-semirings have been introduced and study the properties related to them. Let μ be a fuzzy k-ideal of Γ-semiring M with |Im(μ)| = 2 and μ(0) = 1. Then we establish that M_μ is a 2-prime ideal of Γ-semiring M if and only if μ is a fuzzy prime ideal of Γ-semiring M.

• Communications of the Korean Mathematical Society
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• v.20 no.3
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• pp.427-436
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• 2005

Let (R, m) be a 2-dimensional regular local ring with algebraically closed residue field R/m. Let K be the quotient field of R and v be a prime divisor of R, i.e., a valuation of K which is birationally dominating R and residually transcendental over R. Zariski showed that there are finitely many simple v-ideals $m=P_0\;{\supset}\;P_1\;{\supset}\;{\cdotS}\;{\supset}\;P_t=P$ and all the other v-ideals are uniquely factored into a product of those simple ones. It then was also shown by Lipman that the predecessor of the smallest simple v-ideal P is either simple (P is free) or the product of two simple v-ideals (P is satellite), that the sequence of v-ideals between the maximal ideal and the smallest simple v-ideal P is saturated, and that the v-value of the maximal ideal is the m-adic order of P. Let m = (x, y) and denote the v-value difference |v(x) - v(y)| by $n_v$. In this paper, if the m-adic order of P is 2, we show that $O(P_i)\;=\;1\;for\;1\;{\leq}\;i\; {\leq}\;{\lceil}\;{\frac{b+1}{2}}{\rceil}\;and\;O(P_i)\;=2\;for\;{\lceil}\;\frac{b+3}{2}\rceil\;{\leq}\;i\;\leq\;t,\;where\;b=n_v$. We also show that $n_w\;=\;n_v$ when w is the prime divisor associated to a simple v-ideal $Q\;{\supset}\;P$ of order 2 and that w(R) = v(R) as well.