(c) Show that the ideal generated by x² + y² + z² € C[x, y, z] is a prime ideal.
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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Solve c part only in 30 min and take a thumb up plz
![5.
In this question R is a commutative ring, assumed to be nontrivial (i.e. 01).
An ideal I in R is called a prime ideal if: (i) IR and (ii) whenever a ER and be R
are such that ab € I, then either a € I or be I.
(a) Show that R is an integral domain if and only if the zero ideal {0} is a prime
ideal.
(b)
Suppose that p ER and suppose p0. Show that the principal ideal pR is a prime
ideal if and only if p is prime.
(c) Show that the ideal generated by x² + y² + 2² € C[x, y, z] is a prime ideal.
(d)
For each of the following statements, say whether or not it is always true. Justify
your answer fully.
(i) If I and J are prime ideals in R then In J is a prime ideal in R.
(ii) If I and J are prime ideals in R then IJ is a prime ideal in R.
(iii)
If I and J are prime ideals in R then either I + J = R or I + J is a prime ideal
in R.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F032390c4-bf45-4810-9d69-c6556f2567b0%2F3894364a-8183-4828-89dd-4c4c429d2c4d%2F38vy1t7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5.
In this question R is a commutative ring, assumed to be nontrivial (i.e. 01).
An ideal I in R is called a prime ideal if: (i) IR and (ii) whenever a ER and be R
are such that ab € I, then either a € I or be I.
(a) Show that R is an integral domain if and only if the zero ideal {0} is a prime
ideal.
(b)
Suppose that p ER and suppose p0. Show that the principal ideal pR is a prime
ideal if and only if p is prime.
(c) Show that the ideal generated by x² + y² + 2² € C[x, y, z] is a prime ideal.
(d)
For each of the following statements, say whether or not it is always true. Justify
your answer fully.
(i) If I and J are prime ideals in R then In J is a prime ideal in R.
(ii) If I and J are prime ideals in R then IJ is a prime ideal in R.
(iii)
If I and J are prime ideals in R then either I + J = R or I + J is a prime ideal
in R.
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