Question

Let$R$and$S$ be rings with identity. What are the units in the ring$R\times S$ ?

Short answer

The units in the ring $R\times S$will be the Cartesian product of the elements of individual rings$R$and$S$ .

Step-by-step solution

Elements of Rings

If a ringhas an element $k$, such that$k\in R$ .

Then the following equation will have a unique solutionas:

$k+x=0_R$

Proof

The given rings are$R$and$S$ with the sets of units $R^{\text{'}}$and$S^{\text{'}}$ , respectively.

Let one of the elements in the ring $R\times S$be$\left(r,s\right)$ .

Now, we have:

$$\begin{gathered} \left(r,s\right)\in {\left(R\times S\right)}^{\text{'}} \\ \Rightarrow \left(r,s\right)\left(x,y\right)=\left(1_R,1_S\right)=\left(x,y\right)\left(r,s\right)\left\{\left(x,y\right)\in R\times S\right\} \\ \Rightarrow \left(rx,sy\right)=\left(1_R,1_S\right)=\left(xr,ys\right) \\ \Rightarrow rx=1_R=xrandsy=1_s=ys \\ \Rightarrow r\in R^{\text{'}}ands\in S \\ \Rightarrow \left(r,s\right)\in R^{\text{'}}\times S^{\text{'}} \\ \Rightarrow R\times S=R^{\text{'}}\times S^{\text{'}} \end{gathered}$$

Hence it is proved$R\times S=R^{\text{'}}\times S^{\text{'}}$ .