Prime ideals

Prime ideals

Elementary Algebra

Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018

If R is a commutative ring with unity, thenevery maximal ideal is a prime ideal.R is a field if and only if the only ideals of R are R and {0}.I is a maximal ideal if and only if R/I is a field.

On the Primary Decomposition of k-Ideals and Fuzzy k-Ideals in Semirings

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Journal Information

Published in Fuzzy Information and Engineering, 2021

Ram Parkash Sharma, Madhu Dadhwal, Richa Sharma, S. Kar

(ii) Let for some fuzzy ideals and of R. Then by the definition of fuzzy colon ideal, . Since μ is a fuzzy primary ideal of R and , we have which further implies that either or , as is a fuzzy prime ideal of R.

Ideals of an EMV-semiring

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Published in International Journal of General Systems, 2020

R. A. Borzooei, M. Shenavaei, A. Di Nola, O. Zahiri

We know that each EMV-ideal of is a lattice ideal, thus it is an ideal of S. However, a prime ideal of is not a prime ideal of S, in general. For example, is a prime ideal of EMV-algebra , but it is not a prime ideal of EMV-semiring , since , but .

MVW-rigs and product MV-algebras

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Published in Journal of Applied Non-Classical Logics, 2018

Alejandro Estrada, Yuri A. Poveda

A subset P of an MV-algebra A is an MV-prime ideal if it is an ideal of MV-algebra, and given , implies or .

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Elementary Algebra

On the Primary Decomposition of k-Ideals and Fuzzy k-Ideals in Semirings

Ideals of an EMV-semiring

MVW-rigs and product MV-algebras