AARP14

Prove that the center $Z\,$ of a ring $R\,$ is a subring containing the identity.

The subring criteria for $Z\,$ is $Z\ne \empty\,$ and $\forall a,b \isin Z, a-b\isin Z, ab\isin Z\,$ .

$1r=r1=r\forall r\isin R\,$ so $1\isin Z\,$ and $Z\ne \empty\,$ .

For some $a,b\isin Z, (a-b)r=(ar)-(br)=(ra)-(rb)=r(a-b)\forall r\isin R\,$ so $a-b\isin Z\,$ .

Also $(ab)r=a(br)=a(rb)=(ar)b=(ra)b=r(ab)\,$ so $ab\isin Z\,$ .