Let R be a commutative ring with unit element .if f(x) is a prime ideal of R[x] then show that R is an integral domain.

Amari Flowers

Amari Flowers

Answered question

2021-03-07

Let R be a commutative ring with unit element .if f(x) is a prime ideal of R[x] then show that R is an integral domain.

Answer & Explanation

lamusesamuset

lamusesamuset

Skilled2021-03-08Added 93 answers

Given R be a commutative ring with unit element.
If f(x) is a prime ideal of R[x] then we have to show that R is an integral domain.
That is we have to prove R=R[x]x is an integral domain.
Now, take non zero element in R[x]x say f(x)+(x)andg(x)+(x)
Now,
(f(x)+(x))(g(x)+(x))=(x)
f(x)g(x)+(x)=(x)
f(x)g(x)(x)
Since, (x) is assumed to be a prime ideal
Hence, we have f(x)(x) or g(x)(x)
This implies that either f(x)+(x)=(x) or g(x)+(x)=(x)
Which roves that R=R[x]x is an integral domain.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Commutative Algebra

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?