Maximal ideals

Maximal 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.

Two notions of MV-algebraic semisimplicity relative to fixed MV-chains

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Journal Information

Published in Journal of Applied Non-Classical Logics, 2022

Celestin Lele, Jean B. Nganou, Jean M. Wagoum

Note that every -maximal ideal is prime. Indeed, if M is -maximal, then for some . It follows that A/M is isomorphic to a sub-MV-algebra of . Since is an MV-chain, then A/M is an MV-chain, which implies that M is prime. In addition it is clear from the definition that a proper ideal M of A is -maximal if and only if A/M is isomorphic to a sub-MV-algebra of .

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

Let I be a maximal ideal of S and . If , then . Now, let . Then is a proper ideal of the MV-semiring . If , then by the assumption, and so by Proposition 3.7, a = z + s.x for some and some . Clearly, and so , which implies that is a maximal ideal of .

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Elementary Algebra

Two notions of MV-algebraic semisimplicity relative to fixed MV-chains

Ideals of an EMV-semiring