• Bulletin of the Korean Mathematical Society
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• v.20 no.1
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• pp.45-49
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• 1983

If R is ring and M is a right (or left) R-module, then M is called a faithful R-module if, for some a in R, x.a=0 for all x.mem.M then a=0. In [4], R.E. Johnson defines that M is a prime module if every non-zero submodule of M is faithful. Let us define that M is of prime type provided that M is faithful if and only if every non-zero submodule is faithful. We call a right (left) ideal I of R is of prime type if R/I is of prime type as a R-module. This is equivalent to the condition that if xRy.subeq.I then either x.mem.I ro y.mem.I (see [5:3:1]). It is easy to see that in case R is a commutative ring then a right or left ideal of a prime type is just a prime ideal. We have defined in [5], that a chain of right ideals of prime type in a ring R is a finite strictly increasing sequence I$_{0}$.contnd.I$_{1}$.contnd....contnd.I$_{n}$; the length of the chain is n. By the right dimension of a ring R, which is denoted by dim, R, we mean the supremum of the length of all chains of right ideals of prime type in R. It is an integer .geq.0 or .inf.. The left dimension of R, which is denoted by dim$_{l}$ R is similarly defined. It was shown in [5], that dim$_{r}$R=0 if and only if dim$_{l}$ R=0 if and only if R modulo the prime radical is a strongly regular ring. By "a strongly regular ring", we mean that for every a in R there is x in R such that axa=a=a$^{2}$x. It was also shown that R is a simple ring if and only if every right ideal is of prime type if and only if every left ideal is of prime type. In case, R is a (right or left) primitive ring then dim$_{r}$R=n if and only if dim$_{l}$ R=n if and only if R.iden.D$_{n+1}$ , n+1 by n+1 matrix ring on a division ring D. in this paper, we establish the following results: (1) If R is prime ring and dim$_{r}$R=n then either R is a righe Ore domain such that every non-zero right ideal of a prime type contains a non-zero minimal prime ideal or the classical ring of ritght quotients is isomorphic to m*m matrix ring over a division ring where m.leq.n+1. (b) If R is prime ring and dim$_{r}$R=n then dim$_{l}$ R=n if dim$_{l}$ R=n if dim$_{l}$ R<.inf. (c) Let R be a principal right and left ideal domain. If dim$_{r}$R=1 then R is an unique factorization domain.TEX>R=1 then R is an unique factorization domain.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2005.11a
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• pp.492-495
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• 2005

We give the characterization of an intuitionistic fuzzy ideal[resp. intuitionistic fuzzy left ideal, an intuitionistic fuzzy right ideal and an intuitionistic fuzzy hi-ideal] generated by an intuitionistic fuzzy set in a semigroup without any condition. And we prove that every intuitionistic fuzzy ideal of a semigroup S is the union of a family of intuitionistic fuzzy principle ideals of 5. Finally, we investigate the intuitionistic fuzzy ideal generated by an intuitionistic fuzzy set in $S^{1}$

• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1713-1726
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• 2018

We study the one-sided regularity of matrices in upper triangular matrix rings in relation with the structure of diagonal entries. We next consider a ring theoretic condition that ab being regular implies ba being also regular for elements a, b in a given ring. Rings with such a condition are said to be commutative at regular product (simply, CRP rings). CRP rings are shown to be contained in the class of directly finite rings, and we prove that if R is a directly finite ring that satisfies the descending chain condition for principal right ideals or principal left ideals, then R is CRP. We obtain in particular that the upper triangular matrix rings over commutative rings are CRP.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1887-1903
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• 2013

We continue the study of McCoy condition to analyze zero-dividing polynomials for the constant annihilators in the ideals generated by the coefficients. In the process we introduce the concept of ideal-${\pi}$-McCoy rings, extending known results related to McCoy condition. It is shown that the class of ideal-${\pi}$-McCoy rings contains both strongly McCoy rings whose non-regular polynomials are nilpotent and 2-primal rings. We also investigate relations between the ideal-${\pi}$-McCoy property and other standard ring theoretic properties. Moreover we extend the class of ideal-${\pi}$-McCoy rings by examining various sorts of ordinary ring extensions.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.12 no.3
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• pp.215-225
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• 2012

We initiate the study of interval-valued fuzzy quasi-ideal of a semigroup. In Section 2, we list some basic definitions in the later sections. In Section 3, we investigate interval-valued fuzzy subsemigroups and in Section 4, we define interval-valued fuzzy quasi-ideals and establish some of their basic properties. In Section 5, we obtain characterizations of regular and intraregular semigroups using the machinery developed in the preceding sections.

• 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 $J_2^r(S)=S{\cap}J_2^r(R)$.

• Korean Journal of Mathematics
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• v.22 no.2
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• pp.217-234
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• 2014

Mason extended the reflexive property for subgroups to right ideals, and examined various connections between these and related concepts. A ring was usually called reflexive if the zero ideal satisfies the reflexive property. We here study this property skewed by ring endomorphisms, introducing the concept of an ${\alpha}$-skew reflexive ring, where is an endomorphism of a given ring.

• Korean Journal of Mathematics
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• v.26 no.1
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• pp.143-153
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• 2018

In this paper, we define 0-minimal (m, n)-ideals in an LA-semigroup ${\mathcal{S}}$ and prove that if R(L) is a 0-minimal right (left) ideal of S, then either $R^mL^n=\{0\}$ or $R^mL^n$ is a 0-minimal (m, n)-ideal of S for $m,n{\geq}3$.

• Journal of applied mathematics & informatics
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• v.19 no.1_2
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• pp.551-572
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• 2005

Recently, there are some empirical Bayes procedures using NA samples. We point out a key equality which may not hold for NA samples. Thus, the results of those empirical Bayes procedures based on NA samples are dubious

• Kyungpook Mathematical Journal
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• v.53 no.3
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• pp.435-457
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• 2013

Using the Lukasiewicz 3-valued implication operator, the notion of an (${\alpha},{\beta}$)-intuitionistic fuzzy left (right) $h$-ideal of a hemiring is introduced, where ${\alpha},{\beta}{\in}\{{\in},q,{\in}{\wedge}q,{\in}{\vee}q\}$. We define intuitionistic fuzzy left (right) $h$-ideal with thresholds ($s,t$) of a hemiring R and investigate their various properties. We characterize intuitionistic fuzzy left (right) $h$-ideal with thresholds ($s,t$) and (${\alpha},{\beta}$)-intuitionistic fuzzy left (right) $h$-ideal of a hemiring R by its level sets. We establish that an intuitionistic fuzzy set A of a hemiring R is a (${\in},{\in}$) (or (${\in},{\in}{\vee}q$) or (${\in}{\wedge}q,{\in}$)-intuitionistic fuzzy left (right) $h$-ideal of R if and only if A is an intuitionistic fuzzy left (right) $h$-ideal with thresholds (0, 1) (or (0, 0.5) or (0.5, 1)) of R respectively. It is also shown that A is a (${\in},{\in}$) (or (${\in},{\in}{\vee}q$) or (${\in}{\wedge}q,{\in}$))-intuitionistic fuzzy left (right) $h$-ideal if and only if for any $p{\in}$ (0, 1] (or $p{\in}$ (0, 0.5] or $p{\in}$ (0.5, 1] ), $A_p$ is a fuzzy left (right) $h$-ideal. Finally, we prove that an intuitionistic fuzzy set A of a hemiring R is an intuitionistic fuzzy left (right) $h$-ideal with thresholds ($s,t$) of R if and only if for any $p{\in}(s,t]$, the cut set $A_p$ is a fuzzy left (right) $h$-ideal of R.