Proof: Characteristic of Commutative Ring R[x] is Same as R

[stihl29]

Let R be a commutative ring. Show that the characteristic or R[x] is the same as the characteristic of R.

I'm really not sure where to start on this at all. I'm not sure what is ment by R.

 

[Dick]

R is a commutative ring, they said that. R[x] is the ring of polynomials in x over R. Now at least show some attempt or thought about the problem.

 

[Petek]

Let R be a commutative ring. Show that the characteristic or R[x] is the same as the characteristic of R.

I'm really not sure where to start on this at all. I'm not sure what is ment by R.

I think that there are several typos in your post. Did you mean to state "Show that the characteristic of R[x] is ...". Also, R is the ring in question. Did you mean that you're not sure what is meant by R[x]? If so, R[x] is the ring of polynomials in one variable with coefficients in R. If my assumptions are correct, what is the unit element of R[x]? How does this relate to the definition of the characteristic of the ring?

 

[stihl29]

i need to show for that for a polynomial in say, z mod m the characteristic is m, meaning 1+1+1... (n-summations)

 

[HallsofIvy]

No, you don't because you are NOT dealing with "say, z mod m" you are dealing with an abstract commutative ring. What is the DEFINITION of "characteristic" for a commutat9ive ring? What is the definition of "characteristic" for a ring of polynomials with coefficients in a commutative ring?