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

Question
Commutative Algebra
asked 2020-12-21
Let R be a commutative ring. Show that R[x] has a subring isomorphic to R.

Answers (1)

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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