• Journal of applied mathematics & informatics
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• 제8권3호
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• pp.855-867
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• 2001

This note we give a construction of a quotient ring $R/{\mu}$ induced via a fuzzy ideal ${\mu}$ in a ring R. The Fuzzy First, Second and Third Isomorphism Theorems are established. For some applications of this construction of quotient rings, we show that if ${\mu}$ is a fuzzy ideal of a commutative ring R, then $\mu$ is prime (resp. $R/{\mu}$ is a field, every zero divisor in $R/{\mu}$ is nilpotent). Moreover we give a simpler characterization of fuzzy maximal ideal of a ring.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제7권1호
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• pp.33-40
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• 2000

We prove some characterization of rings with chain conditions in terms of fuzzy quotient rings and fuzzy ideals. We also show that a ring R is left Artinian if and only of the set of values of every fuzzy ideal on R is upper well-ordered.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제13권3호
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• pp.167-187
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• 2006

We introduce the concepts of intuitionistic fuzzy submodules and intuitionistic fuzzy weak congruences on an R-module (Near-ring module). And we obtain the correspondence between intuitionistic fuzzy weak congruences and intuitionistic fuzzy submodules of an R-module. Also, we define intuitionistic fuzzy quotient R-module of an R-module over an intuitionistic fuzzy submodule and obtain the correspondence between intuitionistic fuzzy weak congruences on an R-module and intuitionistic fuzzy weak congruences on intuitionistic fuzzy quotient R-module over an intuitionistic fuzzy submodule of an R-module.

• 호남수학학술지
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• 제42권2호
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• pp.219-234
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• 2020

The fundamental relation on a fuzzy hyperring is defined as the smallest equivalence relation, such that the quotient would be the ring, that is not commutative necessarily. In this paper, we introduce a new fuzzy strongly regular equivalence on fuzzy hyperrings, where the ring is commutative with respect to both sum and product. With considering this relation on fuzzy hyperring, the set of the quotient is a commutative ring. Also, we introduce fundamental functor between the category of fuzzy hyperrings and category of commutative rings and some related properties. Eventually, we introduce α-part in fuzzy hyperring and determine some necessary and sufficient conditions so that the relation α is transitive.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제11권2호
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• pp.127-132
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• 2004

$H_v$-rings first were introduced by Vougiouklis in 1990. The largest class of algebraic systems satisfying ring-like axioms is the $H_v$-ring. Let R be an $H_v$-ring and ${\gamma}_R$ the smallest equivalence relation on R such that the quotient $R/{\gamma}_R$, the set of all equivalence classes, is a ring. In this case $R/{\gamma}_R$ is called the fundamental ring. In this short communication, we study the fundamental rings with respect to the product of two fuzzy subsets.