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Math Help - Derivation and Jordan derivation

  1. #1
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    Derivation and Jordan derivation

    Let R be ring, and d:R \toR 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?)
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  2. #2
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    Quote Originally Posted by Shurelia View Post
    Let R be ring, and d:R \toR 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.
    Last edited by NonCommAlg; May 8th 2011 at 10:41 PM.
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