• 대한수학회보
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• 제28권2호
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• pp.185-190
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• 1991

The concept of "almost invertibility modulo closed ideals" in L(X, Y) was introduced, implicitly, by Robin Harte [1, Theorem 3.9.5]. The aims of this note are to make the formal definition of "almost left and right invertible modulo closed ideals" operators and show that they form open sets.they form open sets.

• 한국지능시스템학회논문지
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• 제17권1호
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• pp.118-122
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• 2007

In this paper, we consider the intuitionistic fuzzification of the notion of a interior ideal in ordered semigroup S, and investigate some properties of such ideals. In terms of intuitionistic fuzzy set, characterizations of intuitionistic fuzzy interior ideals in ordered semigroups are discussed. Using a collection of interior ideals with additional conditions, an intuitionistic fuzzy interiror ideal is constructed. Natural equivalence relations on the set of all intuitionistic fuzzy interior ideals of an ordered semigroup are investigated. We also give a characterization of a intuitionistic fuzzy simple semigroup in terms of intuitionistic fuzzy interior ideals.

• Journal of applied mathematics & informatics
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• 제28권5_6호
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• pp.1217-1225
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• 2010

In this paper we define fuzzy interior $\Gamma$-ideals in ordered $\Gamma$-semigroups. We prove that in regular(resp. intra-regular) ordered $\Gamma$-semigroups the concepts of fuzzy interior $\Gamma$-ideals and fuzzy $\Gamma$-ideals coincide. We prove that an ordered $\Gamma$-semigroup is fuzzy simple if and only if every fuzzy interior $\Gamma$-ideal is a constant function. We characterize intra-regular ordered $\Gamma$-semigroups in terms of interior (resp. fuzzy interior) $\Gamma$-ideals.

• 대한수학회보
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• 제35권4호
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• pp.615-624
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• 1998

We consider a generalization of [1. theorem 2.5]. We give a description of the element of $_{\mathcal I}^l$(resp. $_{\mathcal I} ^r$), where A and B are left (resp. right)${\mathcal I}$-ideals of an IS-algebra X. For a nonempty left (resp. right) stable, we obtain a condition for $_{\mathcal I}^l$(resp. $_{\mathcal I}^l$) to be closed. We give a characterization of a closed $\mathcal I$-ideal in an IS-algebra, and show that in a finite IS-algebra, every $\mathcal I$-ideal is closed.

• 대한수학회지
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• 제51권3호
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• pp.495-507
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• 2014

Nielsen and Rege-Chhawchharia called a ring R right McCoy if given nonzero polynomials f(x), g(x) over R with f(x)g(x) = 0, there exists a nonzero element r ${\in}$ R with f(x)r = 0. Hong et al. called a ring R strongly right McCoy if given nonzero polynomials f(x), g(x) over R with f(x)g(x) = 0, f(x)r = 0 for some nonzero r in the right ideal of R generated by the coefficients of g(x). Subsequently, Kim et al. observed similar conditions on linear polynomials by finding nonzero r's in various kinds of one-sided ideals generated by coefficients. But almost all results obtained by Kim et al. are concerned with the case of products of linear polynomials. In this paper we examine the nonzero annihilators in the products of general polynomials.

• 충청수학회지
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• 제13권2호
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• pp.9-17
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• 2001

A ring R is called right mininjective if every isomorphsim between simple right ideals is given by left multiplication by an element of R. In this paper we consider that the necessary and sufficient condition for that Trivial extension of R by V, i.e. T(R; V ) is mininjective. We also study the split null extension R and S by V.

• 대한수학회보
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• 제20권1호
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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.

• Kyungpook Mathematical Journal
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• 제53권3호
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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.

• Korean Journal of Mathematics
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• 제22권2호
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• pp.367-381
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• 2014

The notion of ${\Gamma}$-semigroups was introduced by Sen in 1981 and that of fuzzy sets by Zadeh in 1965. Any semigroup can be reduced to a ${\Gamma}$-semigroup but a ${\Gamma}$-semigroup does not necessarily reduce to a semigroup. In this paper, we study the coincidences of fuzzy generalized bi-ideals, fuzzy bi-ideals, fuzzy interior ideals and fuzzy ideals in regular, left regular, right regular, intra-regular, semisimple ordered ${\Gamma}$-semigroups.

• Kyungpook Mathematical Journal
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• 제52권4호
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• pp.375-381
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• 2012

We introduce the concepts of ordered quasi-ideals, ordered bi-ideals in an ordered ternary semigroup and study their properties. Also regular ordered ternary semigroup is defined and several ideal-theoretical characterizations of the regular ordered ternary semigroups are furnished.