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We are given that -

1) R is a commutative ring with unity.
2) I is a prime ideal of R.

We have to show that I[x] is a prime ideal of R[x].

MY PROGRESS SO FAR

My progress, so far, is simply that I multiplied two arbitrary polynomial from R[x] that give an arbitrary polynomial of I[x] and tried to show that since ALL the constants belong to I, somehow, since the constants are formed by product, i.e, either of the constants must belong to I.

For example

last constant = ab We must have either a in I or b in I Then I did a similar thing to the second last constant. However, I could never say with guarantee that one particular side belongs to I.

user26857
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Lelouch
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1 Answers1

7

$$R[x]/I[x]\cong (R/I)[x]$$
Since $R/I$ is a domain, $(R/I)[x]$ is a domain, so $I[x]$ is a prime ideal.

user26857
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