Question

Show that$R*=\left\{\left[a,1_R\right]/a\in R\right\}$ is a subring of F.

Short answer

It is proved that R* is a subring of F .

Step-by-step solution

Definition of Subring

Let R be a ring. A non-empty subset S of R is said to be a subring of R if

(i) $a,b\in S\Rightarrow a+\left(-b\right)\in S$

(ii) $a,b\in S\Rightarrow ab\in S$

Proving that is a subring of F

Let $\left[a,1_R\right],\left[b,1_R\right]\in R*$.

Find $\left[a,1_R\right]+\left[-b,1_R\right]$as:

$\begin{array}{rcl}\left[a,1_R\right]+\left[-b,1_R\right]& =& \left[a{1}_R+1_R\left(-b\right),1_R{1}_R\right]\\ & =& \left[a-b,1_R\right]\in R*\left[\because a,b\in R\Rightarrow a-b\in R\right]\end{array}$

Now, find $\left[a,1_R\right]\left[b,1_R\right]$as:

$\begin{array}{rcl}\left[a,1_R\right]\left[b,1_R\right]& =& \left[ab,1_R,1_R\right]\\ & =& \left[ab,1_R\right]\in R*[\because a,b\in R\Rightarrow ab\in R*]\end{array}$

Therefore, R*is a subring of F .