Understanding Homomorphisms: The Relationship Between A and B

[tgt]

Suppose there exists a homomorphism f:A->B then does it make sense to have

$$f_{Ker(f)}:Ker(f) \to B$$ ?

why doesn't my tex show up?

Offcourse, $$Im(f_{Ker(f)})=1_{B}$$

Moderator Note: [/color]Fixed LaTeX.

 

[Hootenanny]

Suppose there exists a homomorphism f:A->B then does it make sense to have

why doesn't my tex show up?

Offcourse, Im(f_{Ker(f)})=1_{B}

You need to close the tex enviroment. e.g.

[tex]\frac{dy}{dx}[ /tex]

(without the space in the square brackets of course)

 

[morphism]

What are you trying to say?

 

[daveyinaz]

Are you trying to say that the elements in A that are in the kernel map into B. Well yes that's true since they all map to the zero element within B.

 

[tgt]

What are you trying to say?

Given a map, we can define a new map, mapping the kernel of the map to the range of the map.

 

[morphism]

Yes, but what's the point?

 

[HallsofIvy]

Suppose there exists a homomorphism f:A->B then does it make sense to have

$$f_{Ker(f)}:Ker(f) \to B$$ ?

why doesn't my tex show up?

Offcourse, $$Im(f_{Ker(f)})=1_{B}$$

Moderator Note: [/color]Fixed LaTeX.

Yes, that makes sense. It isn't very interesting, however, since $f_{Ker(f)}$ is just the identity map f(x)= 1_B, as you say.