# Thread: Multiplication by scalar in K[X]/(f)

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

Hi:
Let K[X] be the ring of polynomials over the field K and let
f $\displaystyle \epsilon$ 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?

2. Originally Posted by ENRIQUESTEFANINI
Hi:
Let K[X] be the ring of polynomials over the field K and let
f $\displaystyle \epsilon$ 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?
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

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.

4. Originally Posted by ENRIQUESTEFANINI
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

5. Originally Posted by tonio
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.