# Thread: Center as an ideal....

1. ## Center as an ideal....

Hello all. How can I show that the center of a ring Z(R) is an ideal of a ring R iff R is commutative?

Thanks!

2. Why don't you start with the easier part of the proof. Start with "If $R$ is commutative, then $Z(R)$ is an ideal in $R$."

3. Well, the center is every element in the ring that commutes with the whole ring. So if the whole ring is commutative, the center will be the whole ring. So Z(R)=R, so Z(R) is an ideal....the other way of the if and only if proof is what I have been struggling with...

4. Originally Posted by smacktalk88
Well, the center is every element in the ring that commutes with the whole ring. So if the whole ring is commutative, the center will be the whole ring. So Z(R)=R, so Z(R) is an ideal....the other way of the if and only if proof is what I have been struggling with...
Hint - Under the assumption Z(R) is ideal - show for every 'r' in R; 'r' belongs to Z(R)

This is not all that tough, I guess. Just use the definition.

5. Ahhhh. Thanks a lot. This should help.

6. Originally Posted by smacktalk88
Hello all. How can I show that the center of a ring Z(R) is an ideal of a ring R iff R is commutative?

Thanks!

If $I\subseteq R$ is an ideal and $1\in I$ then obviously $I=R$ ... apply this to $I= Z(R)$