Question

If R is commutative, show that R[x] is also commutative.

Short answer

It is proved that R[x] is commutative.

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(x)+g(x) exists.

Commutative Polynomials

The multiplication for any polynomials with elements such that $a,b\in R$, can be expressed as:

localid="1654010940705" $\begin{array}{rcl}\left\{\underset{i=0}{\overset{m}{\sum a_i}}{x}^i\right\}\left\{\underset{j=0}{\overset{n}{\sum b_j}}{x}^j\right\}& =& \sum _{k=0}^{m+n}\left(\sum _{i+j=k}{a}_i{b}_i\right)x^k\\ & =& \sum _{k=0}^{m+n}\left({\sum _{i+j=k}}_{}{b}_i{a}_i\right)x^k\\ & =& \left\{\sum _{j=0}^n{b}_j{x}^j\right\}\left\{\sum _{i=0}^m{a}_i{x}^i\right\}\end{array}$

Clearly, the subsets are commutative.

Hence proved, if R is commutative then, R[x] is also commutative.