# Derivation and Jordan derivation

• May 8th 2011, 07:00 AM
Shurelia
Derivation and Jordan derivation
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?)
• May 8th 2011, 09:22 PM
NonCommAlg
Quote:

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.