Let be a ring. Recall that an additive map is called a derivation if for all Thus if is a derivation of then for all

**Definition**. Let be a ring. An additive map is called a **Jordan derivation** if for all

So every derivation is a Jordan derivation . But the converse is not true and this is not surprising:

**Example**. Let with the relation Let which is an ideal of because Let

See that with matrix addition and multiplication, is a ring because is an ideal of For any define Then is a Jordan derivation but not a derivation.

*Proof*. It is obvious that is additive. Let Then, since we have

and so is a Jordan derivation. To see why is not a derivation, let and Then but

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