Finitely generated ring.

Question

Let $R$ be a finitely generated ring. Then is it true that $R^2$ is also finitely generated?

My Attempt: I do not find a counterexample. I think it is true. Please someone help me to prove that $R^2$ is finitely generated.

Answer 1

Let $A\subseteq R$ be a set such that no proper subring of $R$ contains $A$. Then $B:=\{\,(a,1)\mid a\in A\,\}\cup\{\,(1,a)\mid a\in A\,\}$ is a subset of the ring $R^2$ having the same property (I assume $1\in R$). Note that if $A$ is finite than $B$ is also finite.