DEFINITION: We define an ideal I C R to be a prime ideal if it is a proper ideal of R with the property that whenever ab e I we have either a e I or b € I. CLAIM: Every maximal ideal J of R is a prime ideal. Use the second claim and the result from homework, Z[x]/(x) = Z, to argue that not every prime ideal is maximal.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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DEFINITION: We define an ideal I C R to be a prime ideal if it is a proper ideal of R with the
property that whenever ab e I we have either a e I or b € I.
CLAIM:
Every maximal ideal J of R is a prime ideal. Use the second claim and the result
from homework, Z[x]/(x) = Z, to argue that not every prime ideal is maximal.
Transcribed Image Text:DEFINITION: We define an ideal I C R to be a prime ideal if it is a proper ideal of R with the property that whenever ab e I we have either a e I or b € I. CLAIM: Every maximal ideal J of R is a prime ideal. Use the second claim and the result from homework, Z[x]/(x) = Z, to argue that not every prime ideal is maximal.
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