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Math Help - Multiplication by scalar in K[X]/(f)

  1. #1
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    Multiplication by scalar in K[X]/(f)

    Hi:
    Let K[X] be the ring of polynomials over the field K and let
    f  \epsilon K[X]. Let K[X]/(f) be the quotient
    ring by (f). In K[X]/(f) how can I define multiplication by
    a scalar in K in order to make K[X]/(f) into a K-vector space?
    Thanks for reading.
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  2. #2
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    Quote Originally Posted by ENRIQUESTEFANINI View Post
    Hi:
    Let K[X] be the ring of polynomials over the field K and let
    f  \epsilon K[X]. Let K[X]/(f) be the quotient
    ring by (f). In K[X]/(f) how can I define multiplication by
    a scalar in K in order to make K[X]/(f) into a K-vector space?
    Thanks for reading.
    If g(x) is in K[x] and k is in K, simply define k*(g(x) + (f)):= kg(x) + (f). Is easy, though slightly annoying, to check this is well defined.

    Tonio
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  3. #3
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    Thank you for your reply.
    I see. And I also see that, with this multiplication, K[X]/(f)
    is a K-algebra. My question realy is: is this the only way to define a multiplication by scalar
    that makes K[X]/(f) into a K-algebra? Or is it that there are more than one way of building
    an K-algebra out of K[X]/(f)?

    Enrique.
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  4. #4
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    Quote Originally Posted by ENRIQUESTEFANINI View Post
    Thank you for your reply.
    I see. And I also see that, with this multiplication, K[X]/(f)
    is a K-algebra. My question realy is: is this the only way to define a multiplication by scalar
    that makes K[X]/(f) into a K-algebra? Or is it that there are more than one way of building
    an K-algebra out of K[X]/(f)?

    Enrique.

    Not sure though definitely the above is the most "natural" way since it comes from the definition of scalar multiplication that makes K[x] a K-algebra.

    Tonio
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  5. #5
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    Quote Originally Posted by tonio View Post
    Not sure though definitely the above is the most "natural" way since it comes from the definition of scalar multiplication that makes K[x] a K-algebra.

    Tonio
    Thanks.
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