Question

If R has multiplicative identity $1_R$, show that $1_R$ is also the multiplicative index of R[x] .

Short answer

It is proved that $1_R$ is the multiplicative index of 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(x)+g(x) exists.

Distributive Polynomials

It is given that, $1_R$ is the multiplicative index of R. Now, using distributive property, we have, $\forall a\in R$:

$1_R\left\{\sum _{i=0}^m{a}_i{x}^i\right\}=\underset{i=0}{\overset{m}{\sum 1_R}}{a}_i{x}^i=\sum _{i=0}^m{a}_i{x}^i$

$\left\{\sum _{i=0}^m{a}_i{x}^i\right\}{1}_R=\sum _{i=0}^m{a}_i{x}^i{1}_R=\sum _{i=0}^m{a}_i{1}_R{x}^i\underset{i=0}{\overset{m}{=\sum }}{a}_i{x}^i$

Clearly, the subset satisfies the distributive property.

Hence proved, $1_R$ is multiplicative index of R[x].