To find unity of a subring of ring R.

Question

"Given a ring R, let S=Z×R. show that S is a ring with unity under coordinatewise addition but the multiplication being $(m,n)(n,y)=(mn, my+nx+xy)$.

Identifying R with the subset T=${ {(0,x)/ x \in R} }$ of S, verify that T is a subring of S. Thus any ring R can be realised as a subring of a ring S with unity.

Show that 1 of T is not same as 1 of S if R has 1"

I just completed the first part that is to show that S is a ring . However I am not able to find unity of S. Also for the subset T of S, the unity element I have found is (a,b) such that a+b= $1_R$.

I want to show that unity of T is not same as unity of S. Please help.

Answer 1

If $R$ has $1$, then $(0,1)\in T$.

Compute what $(0,1)(0,r)$ and $(0,r)(0,1)$ are for any $r\in R$. Draw your conclusion.

Finally $(1,0)\neq (0,1)$!