# Center of a ring is a subring that contains identity, but what happens in the case of ring of all Even integers?

The set $(E, +, .)$ of even integers forms a commutative ring without unit element( multiplicative identity), with usual rules of addition and multiplication operation.

Next, the $center$ of a ring R is {${z\in R :zr=rz,\forall r\in R}$} (i,e. the set of all elements which commute with every element of R).

I have problem to prove that the center of a ring is a subring that contains the identity.

Since the ring of even integers is commutative, so every element of this ring commutes with other. So I think that its center should be the ring itself. But how does it contain unit element?

The definition of center of a ring doesn't mention the identity: $$Z(R)=\{z\in R:zr=rz, \forall r\in R\}$$ Closure under subtraction and multiplication is easy to prove.
It is also a fact that, if $R$ has an identity $1$, then $1\in Z(R)$, simply because $$1r=r=r1$$ by definition.
If a ring $R$ (with or without identity) is commutative, then obviously $Z(R)=R$.