Derivation and Jordan derivation

**Shurelia** · May 8th 2011, 08:00 AM

Let R be ring, and d:R $\to$ R is aditif mapping then
d is called derivation if
d(ab)=d(a)b+ad(b) $\forall$ a,b $\in$ R
d is called Jordan derivation if
d(a^2)=d(a)a+ad(a) $\forall$ a $\in$ R
Obviously any derivation is Jordan derivation.

But the converse is not true. (is there any example to show this statement?)

**NonCommAlg** · May 8th 2011, 10:22 PM

Originally Posted by Shurelia

Let R be ring, and d:R $\to$ R is aditif mapping then
d is called derivation if
d(ab)=d(a)b+ad(b) $\forall$ a,b $\in$ R
d is called Jordan derivation if
d(a^2)=d(a)a+ad(a) $\forall$ a $\in$ R
Obviously any derivation is Jordan derivation.

But the converse is not true. (is there any example to show this statement?)

it's a little bit tricky! here is an example: let $S$ be the algebra $\mathbb{C}[x]$ with the relation $x^2=0$ . let $I = \mathbb{C}x$ . note that $I$ is an ideal of $S$ because $x^2=0$ . now let $R$ be the ring of all $2 \times 2$ matrices in the form $r = \begin{pmatrix}a & b \\ c & d \end{pmatrix}$ where $a,b,d \in S$ and $c \in I$ . define the map $\delta : R \longrightarrow R$ by $\delta(r)=\begin{pmatrix} 0 & c \\ 0 & 0 \end{pmatrix}$ . then

1) $\delta(r_1+r_2)=\delta(r_1)+\delta(r_2),$ for all $r_1,r_2 \in R.$
2) $\delta(r^2)=r \delta(r) + \delta(r)r,$ for all $r \in R.$
3) there exist $r_1,r_2 \in R$ such that $\delta(r_1r_2) \neq \delta(r_1)r_2 + r_1 \delta(r_2).$

i'll leave it to you to check that 1), 2) and 3) hold.

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