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.

Question
Commutative Algebra
asked 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.

Answers (1)

2021-03-08
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 \(\displaystyle{R}=\frac{{{R}{\left[{x}\right]}}}{{{x}}}\) is an integral domain.
Now, take non zero element in \(\displaystyle\frac{{{R}{\left[{x}\right]}}}{{{x}}}\) say \(\displaystyle{f{{\left({x}\right)}}}+{\left({x}\right)}{\quad\text{and}\quad}{g{{\left({x}\right)}}}+{\left({x}\right)}\)
Now,
\(\displaystyle{\left({f{{\left({x}\right)}}}+{\left({x}\right)}\right)}{\left({g{{\left({x}\right)}}}+{\left({x}\right)}\right)}={\left({x}\right)}\)
\(\displaystyle\Rightarrow{f{{\left({x}\right)}}}{g{{\left({x}\right)}}}+{\left({x}\right)}={\left({x}\right)}\)
\(\displaystyle\Rightarrow{f{{\left({x}\right)}}}{g{{\left({x}\right)}}}\in{\left({x}\right)}\)
Since, (x) is assumed to be a prime ideal
Hence, we have \(\displaystyle{f{{\left({x}\right)}}}\in{\left({x}\right)}\) or \(\displaystyle{g{{\left({x}\right)}}}\in{\left({x}\right)}\)
This implies that either \(\displaystyle{f{{\left({x}\right)}}}+{\left({x}\right)}={\left({x}\right)}\) or \(\displaystyle{g{{\left({x}\right)}}}+{\left({x}\right)}={\left({x}\right)}\)
Which roves that \(\displaystyle{R}=\frac{{{R}{\left[{x}\right]}}}{{{x}}}\) is an integral domain.
0

Relevant Questions

asked 2020-11-22
Suppose that R and S are commutative rings with unites, Let PSJphiZSK be a ring homomorphism from R onto S and let A be an ideal of S.
a. If A is prime in S, show that \(\displaystyle\phi^{{-{{1}}}}{\left({A}\right)}={\left\lbrace{x}\in{R}{\mid}\phi{\left({x}\right)}\in{A}\right\rbrace}\) is prime \(\displaystyle\in{R}\).
b. If A is maximal in S, show that \(\displaystyle\phi^{{-{{1}}}}{\left({A}\right)}\) is maximal \(\displaystyle\in{R}\).
asked 2020-11-27
Let R be a commutative ring. If I and P are idelas of R with P prime such that \(\displaystyle{I}!\subseteq{P}\), prove that the ideal \(\displaystyle{P}:{I}={P}\)
asked 2020-11-02
Let R be a commutative ring with identity and let I be a proper ideal of R. proe that \(\displaystyle\frac{{R}}{{I}}\) is a commutative ring with identity.
asked 2020-12-17
If R is a commutative ring with unity and A is a proper ideal of R, show that \(\displaystyle\frac{{R}}{{A}}\) is a commutative ring with unity.
asked 2020-12-09
Suppose that R is a commutative ring and |R| = 30. If I is an ideal of R and |I| = 10, prove that I is a maximal ideal.
asked 2020-12-21
Let R be a commutative ring. Show that R[x] has a subring isomorphic to R.
asked 2020-11-29
Give an example of a commutative ring without zero-divisors that is not an integral domain.
asked 2021-03-02
Let R be a commutative ring with unity and a in R. Then \(\displaystyle{\left\langle{a}\right\rangle}={\left\lbrace{r}{a}:{r}\in{R}\right\rbrace}={R}{a}={a}{R}\)
asked 2020-11-23
If R is a commutative ring, show that the characteristic of R[x] is the same as the characteristic of R.
asked 2020-12-09
If A and B are ideals of a commutative ring R with unity and A+B=R show that \(\displaystyle{A}\cap{B}={A}{B}\)
...