If R is a commutative ring, then, the characteristic of R{x} is the same as the characteristic of R.
In ring theory, a commutative ring can be defined as a ring in which the multiplication operation is considered to be commutative in nature.
Let R represent a commutative ring with characteristic K.
Therefore, we have: Kr = 0∀r ∈ R.
Assuming f(x) ∈ R(x)
Then, f(x) would be given by: [tex]a_n x^n + a_{n-1} x^{n-1} +.......+ a_1 x^1 +a_0=0[/tex]
Also, in some instances ai ∈ R and n ∈ N.
Finally, we would have:
[tex]Kf(x) = K(a_n x^n + a_{n-1} x^{n-1} +.......+ a_1 x^1 +a_0)\\\\Kf(x) = Ka_n x^n + Ka_{n-1} x^{n-1} +.......+ Ka_1 x^1 +Ka_0\\\\Kf(x) = 0\times x^n + 0\times x +.......+ 0[/tex]
Kf(x) = 0.
Therefore, we can infer and logically deduce that if R is a commutative ring, then, the characteristic of R{x} is the same as the characteristic of R.
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