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.

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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