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Math Help - Derivation, Higher derivations in ring, basic question, example

  1. #1
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    Derivation, Higher derivations in ring, basic question, example

    this should be definition of higher derivations
    R be ring, N be natural number with {0}
    Let D= { d_{n}},n  \in N be family of additive mappings d_{n}:R \rightarrow R.
    D is said to be higher derivation on R if:
    1. d_{0}=I_{R}
    2. d_{n}(ab)=  \[\sum_{i+j=n}^{.}d_{i}(a) d_{j}(b)\], \foralla,b \inR, and n \in N

    i'm interest in this topics, cause it can be related in calculus.

    if n=1, then its just derivation d_{n}(ab)=d(a)b+ad(b).
    its easy, cause additive mapping represent sum rule, and that condition represent product rule in calculus. (leibniz rule if i'm not mistaken).
    in calculus: the map of d is fixed, d(f(x)) \rightarrowf'(x)
    so, the example is common polynomials
    in algebra: the common example is the map d(x)=ax-xa, a is constan,
    or matrices with little tricks can be derivation

    so the basic is the same, but in algebra we see it at different angle.

    when i learn higher derivations, i have some problems.
    in calculus, d_{2}(ab)=d_{2}(a)(b)+2d(a)d(b)+ad_{2}(b)
    and so on in third,fourth,.... there is constan which follow pascal's triangle, but on definition of higher derivation it ignored.
    so i can't find example in calculus like derivations
    And i still can't find example in algebra which is simple (usually, the example is in general shape, so must be prove with induction, its still too abstract like not as example)

    is there any example of higher derivation, so it can be fun to learn like derivation?

    if my assumption, explanation, definition is wrong, please tell me before i lost any further.
    thanks
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  2. #2
    MHF Contributor

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    Re: Derivation, Higher derivations in ring, basic question, example

    Quote Originally Posted by Misha View Post
    this should be definition of higher derivations
    R be ring, N be natural number with {0}
    Let D= { d_{n}},n  \in N be family of additive mappings d_{n}:R \rightarrow R.
    D is said to be higher derivation on R if:
    1. d_{0}=I_{R}
    2. d_{n}(ab)=  \[\sum_{i+j=n}^{.}d_{i}(a) d_{j}(b)\], \foralla,b \inR, and n \in N

    i'm interest in this topics, cause it can be related in calculus.

    if n=1, then its just derivation d_{n}(ab)=d(a)b+ad(b).
    its easy, cause additive mapping represent sum rule, and that condition represent product rule in calculus. (leibniz rule if i'm not mistaken).
    in calculus: the map of d is fixed, d(f(x)) \rightarrowf'(x)
    so, the example is common polynomials
    in algebra: the common example is the map d(x)=ax-xa, a is constan,
    or matrices with little tricks can be derivation

    so the basic is the same, but in algebra we see it at different angle.

    when i learn higher derivations, i have some problems.
    in calculus, d_{2}(ab)=d_{2}(a)(b)+2d(a)d(b)+ad_{2}(b)
    and so on in third,fourth,.... there is constan which follow pascal's triangle, but on definition of higher derivation it ignored.
    so i can't find example in calculus like derivations
    And i still can't find example in algebra which is simple (usually, the example is in general shape, so must be prove with induction, its still too abstract like not as example)

    is there any example of higher derivation, so it can be fun to learn like derivation?

    if my assumption, explanation, definition is wrong, please tell me before i lost any further.
    thanks
    are you looking for examples of higher derivations in algebra? you mentioned a few of them yourself. so what kind of examples are you looking for? what do you mean by "simple" examples?
    i just mention this point that the set of derivations of a ring is is a ring with addition and the Lie bracket. this fact allows you to construct new derivations from the old ones. so if \delta_1 and \delta_2 are two derivations, so are \delta_1 \pm \delta_2 and [\delta_1,\delta_2]=\delta_1 \delta_2 - \delta_2 \delta_1.
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