Question

If $c\in R$is a zero divisor in a commutative ring R , then c is also a zero divisor in R[x].

Short answer

It is proved that c is a zero divisor in R[x].

Step-by-step solution

Polynomial Arithmetic: 

If any given function R[x] is a ring, then the commutative, associative, and distributive laws hold such that the function $f\left(x\right)+g\left(x\right)$ exists.

Distributive Polynomials

It is given that c is a zero divisor in a commutative ring R.

Then, $\forall a\in R$, we have:

ca=0

ac=0

Since, $R\subseteq R\left[x\right]$for polynomial of 0 degree. Then, $\forall a,c\in R$, we have:

ca=ac=0

Hence proved, is a zero divisor in R[x].