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Which one of the following ideals is prime?

If $T=\mathbb{R}[X,Y,Z]$, then which one is a prime ideal:
A) $\langle X^{2},Y+Z\rangle$
B) $\langle X^{2}YZ,Z\rangle$
C) $\langle XY,XZ+YZ\rangle$
D) $\langle XY,XZ\rangle$.

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marked as duplicate by Rahul, Ittay Weiss, Paul, Davide Giraudo, Stefan Hansen Jan 26 '13 at 10:46

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You should at least tell us what you have tried doing... All of these examples can be handled with absolutely nothing more than the definition of what prime ideal is, really! –  Mariano Suárez-Alvarez Jan 25 '13 at 7:46
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1 Answer

up vote 5 down vote accepted

An ideal $I$ is prime if $ab \in I$ implies $a \in I$ or $b\in I$ for all $a,b \in R$. Some hints:

D) $XY-XZ = X(Y-Z) \in I$ but neither $X$ nor $Y-Z$ in $I$.

C) $XY \in I$ but neither $X$ nor $Y$ in $I$.

A) Similarly, $X^2$ is in $I$ but $X \notin I$.

B) $\langle X^2 ZY \rangle \subseteq \langle Z \rangle$. Therefore $\langle X^2 YZ, Z \rangle = \langle Z \rangle$. And $\langle Z \rangle$ is prime.

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very thanks goobie –  rese Jan 25 '13 at 10:21
    
you are welcome @reme –  goobie Jan 25 '13 at 10:23
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