Adjoining an identity to a ring

Question

I am run into the following in an Algebra text:

"Let $R_0=\mathbb Z/2\mathbb Z⊕\mathbb Z/2\mathbb Z⊕\cdots$ viewed as a ring without identity, with addition and multiplication defined componentwise. Let $R=\mathbb Z⊕R_0$ be the ring obtained by "adjoining" an identity $1\in \mathbb Z$ to $R_0$."

My question is:

What is the identity of $R$?

If it is the pair $(1,0)$ so for any nonzero element $e\in R_0$ we would have $(1,0)(0,e)=(0,e)$ so $(0,0)=(0,e)$ and therefore $e=0$, a contradiction.

Any leading answer would be thanked.

Answer 1

Multiplication is not defined coordinatewise, rather $(n,r)(m,s)=(nm,rs+rm+ns)$. In particular $(1,0)(n,s)=(n,s)$. Writing $(n,r)$ as $n+r$ should help in understanding why we have defined multiplication in such a way.