For an integral domain $R$, the rings $R\times R $ and $R\times R\times R$ are not isomorphic
On contrary suppose that both are isomorphic then if G is prime ideal of one ring then its isomorphic copy must be prime ideal of other
we will construct projection map $p_1:R\times R\to R$ as $p_1(a,b)=a$
Now here kernel is $R$
then by prime ideal theorem $(R\times R )/R\cong R$ which is integral domain so R is prime ideal of R.
Now we will again construct projection map $p_2:R\times R\times R\to R\times R$
Now here kernel is $R$ Now $(R\times R \times R )/R\cong R\times R $ but it is easy to show that $R\times R $ is not integral domain.
So our assumption is wrong .
Hence both are not isomorphic.
Is my argument are valid?
Thanks in advanced.