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

• Oct 17th 2009, 05:40 AM
ENRIQUESTEFANINI
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?
• Oct 17th 2009, 09:59 AM
tonio
Quote:

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
• Oct 17th 2009, 04:50 PM
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.
• Oct 17th 2009, 07:47 PM
tonio
Quote:

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
• Oct 18th 2009, 01:44 AM
ENRIQUESTEFANINI
Quote:

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.