Let R be a commutative ring. Show that R[x] has a subring isomorphic to R.

Question

Question

Answers (1)

Relevant Questions

asked 2021-01-19

Let R be a commutative ring. Prove that \(\displaystyle{H}{o}{m}_{{R}}{\left({R},{M}\right)}\) and M are isomorphic R-modules

See answers (1)

asked 2021-02-11

Let a belong to a ring R. Let \(\displaystyle{S}={\left\lbrace{x}\in{R}{\mid}{a}{x}={0}\right\rbrace}\) . Show that S is a subring of R.

See answers (1)

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 \(\displaystyle\in{R}\).

See answers (1)

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.

See answers (1)

asked 2021-01-27

Assume that the ring R is isomorphic to the ring R'. Prove that if R is commutative, then R' is commutative.

See answers (1)

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 \(\displaystyle{P}:{I}={P}\)

See answers (1)

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.

See answers (1)

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.

See answers (1)

asked 2021-03-02

Let R be a commutative ring with unity and a in R. Then \(\displaystyle{\left\langle{a}\right\rangle}={\left\lbrace{r}{a}:{r}\in{R}\right\rbrace}={R}{a}={a}{R}\)

See answers (1)

asked 2020-12-09

If A and B are ideals of a commutative ring R with unity and A+B=R show that \(\displaystyle{A}\cap{B}={A}{B}\)

See answers (1)

Answer 1

Let R be a commutative rng and consider R[x]
Define: \(\displaystyle\phi:{R}\to{R}{\left[{x}\right]}\) by \(\displaystyle{r}\mapsto{r}\)
Here, clearly we can say that \(\displaystyle\phi\) is one-to-one and homomorphism
Now, \(\displaystyle\phi{\left({R}\right)}\) is subsring of R[x] since it is the imag of a homomorphism
Then \(\displaystyle\phi{\left({R}\right)}\) is subsring of R[x] isomorphic to R

Didn’t find what you are looking for?

Let R be a commutative ring. Show that R[x] has a subring isomorphic to R.

Answers (1)

Relevant Questions