### Abstract

The object of this paper is to study fuzzy ideals over Artinian rings. Among others we prove that a ring with unity is left (right) Artinian if and only if every fuzzy left (right) ideal μ of R is finite valued. This gives a characterization of artinian rings in terms of fuzzy ideals. Mukherjee and Sen (1987) proved that if every fuzzy ideal of a ring with unity is finite valued then the ring is Noetherian. We give an example to show that the converse of this theorem is not true. We also show that certain results of Pan (1987) are not true.

Original language | English |
---|---|

Pages (from-to) | 111-115 |

Number of pages | 5 |

Journal | Fuzzy Sets and Systems |

Volume | 37 |

Issue number | 1 |

DOIs | |

State | Published - Aug 15 1990 |

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### All Science Journal Classification (ASJC) codes

- Artificial Intelligence
- Computer Science Applications
- Computer Vision and Pattern Recognition
- Information Systems and Management
- Statistics, Probability and Uncertainty
- Electrical and Electronic Engineering
- Statistics and Probability

### Cite this

*Fuzzy Sets and Systems*,

*37*(1), 111-115. https://doi.org/10.1016/0165-0114(90)90069-I

**Fuzzy ideals of artinian rings.** / Malik, Davender S.

Research output: Contribution to journal › Article

*Fuzzy Sets and Systems*, vol. 37, no. 1, pp. 111-115. https://doi.org/10.1016/0165-0114(90)90069-I

}

TY - JOUR

T1 - Fuzzy ideals of artinian rings

AU - Malik, Davender S.

PY - 1990/8/15

Y1 - 1990/8/15

N2 - The object of this paper is to study fuzzy ideals over Artinian rings. Among others we prove that a ring with unity is left (right) Artinian if and only if every fuzzy left (right) ideal μ of R is finite valued. This gives a characterization of artinian rings in terms of fuzzy ideals. Mukherjee and Sen (1987) proved that if every fuzzy ideal of a ring with unity is finite valued then the ring is Noetherian. We give an example to show that the converse of this theorem is not true. We also show that certain results of Pan (1987) are not true.

AB - The object of this paper is to study fuzzy ideals over Artinian rings. Among others we prove that a ring with unity is left (right) Artinian if and only if every fuzzy left (right) ideal μ of R is finite valued. This gives a characterization of artinian rings in terms of fuzzy ideals. Mukherjee and Sen (1987) proved that if every fuzzy ideal of a ring with unity is finite valued then the ring is Noetherian. We give an example to show that the converse of this theorem is not true. We also show that certain results of Pan (1987) are not true.

UR - http://www.scopus.com/inward/record.url?scp=38249016991&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249016991&partnerID=8YFLogxK

U2 - 10.1016/0165-0114(90)90069-I

DO - 10.1016/0165-0114(90)90069-I

M3 - Article

AN - SCOPUS:38249016991

VL - 37

SP - 111

EP - 115

JO - Fuzzy Sets and Systems

JF - Fuzzy Sets and Systems

SN - 0165-0114

IS - 1

ER -