**Theorem**. Let be a ring and Let be the center of If is reduced and for all then .

*Proof*. Let Then

So Hence because is reduced. Thus

One class of rings with reduced centers is the class of semiprime rings. If is not reduced, the result in the theorem need not hold. There is a nice example in Lam’s book, “A First Course in Noncommutative Rings”*.* Here it is*:*

**Example**. let be a ring with 1 and let

Let Then for every but obviously because, for example, does not commute with

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