# If R is a commutative ring, show that the characteristic of R[x] is the same as the characteristic of R.

Question
Commutative Algebra
If R is a commutative ring, show that the characteristic of R[x] is the same as the characteristic of R.

2020-11-24
Let R be a commutative ring with characteristic k. Then kr=0 for all $$\displaystyle{r}\in{R}$$. Now, let f(x) in R[x]ZSK.
Let $$\displaystyle{f{{\left({x}\right)}}}={a}_{{n}}{x}^{{n}}+{a}_{{{n}-{1}}}{x}^{{{n}-{1}}}+\ldots+{a}_{{1}}{x}+{a}_{{0}}$$ for some a_i in RZSK
Conider,
$$\displaystyle{k}{f{{\left({x}\right)}}}={\left({k}{a}_{{n}}\right)}{x}^{{n}}+{\left({k}{a}_{{{n}-{1}}}{x}^{{{n}-{1}}}+\ldots+{\left({k}{a}_{{1}}\right)}{x}+{k}{a}_{{0}}\right.}$$
=0+0+...+0
=0
Hence, charecteristic of R[x] is at most k. Since, for all $$\displaystyle{r}\in{R},{r}\in{R}{\left[{x}\right]}$$, the characteristic of R[x] must be at least k. Hence, the characteristic of R[x] is exactly k.

### Relevant Questions

Suppose that R is a commutative ring without zero-divisors. Show that all the nonzero elements of R have the same additive order.
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 $$\displaystyle\in{R}$$.
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.
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.
If A and B are ideals of a commutative ring R with unity and A+B=R show that $$\displaystyle{A}\cap{B}={A}{B}$$
Suppose that R is a ring and that $$\displaystyle{a}^{{2}}={a}$$ for all $$\displaystyle{a}\in{R}{Z}$$. Show that R is commutative.
All idempotent elements in a commutative H ring with a characteristic 2 Check if they are creating a sub-ring. ($$\displaystyle{a}\in{H}$$ idempotent $$\displaystyle\Leftrightarrow{a}^{{2}}={a}$$)
Show that a ring is commutative if it has the property that ab = ca implies b = c when $$\displaystyle{a}\ne{0}$$.