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

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

2020-12-22
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

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