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

Trent Carpenter 2020-11-23 Answered
If R is a commutative ring, show that the characteristic of R[x] is the same as the characteristic of R.
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smallq9
Answered 2020-11-24 Author has 106 answers

Let R be a commutative ring with characteristic k. Then kr=0 for all rR. Now, let f(x)R[x].
Let f(x)=anxn+an1xn1++a1x+a0 for some aiR
Conider,
kf(x)=(kan)xn+(kan1xn1++(ka1)x+ka0
=0+0+...+0
=0
Hence, charecteristic of R[x] is at most k. Since, for all rR,rR[x], the characteristic of R[x] must be at least k. Hence, the characteristic of R[x] is exactly k.

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