Composition of ring homomorphism

Question

I have three rings $A,B,C$ and ring homomorphisms $f: A \rightarrow B$ and $g: B \rightarrow C$, which are both surjective. Is it true that $C$ is isomorphic to $$ A / (\ker(f), \ker(g)) ? $$ If so how can I show this? Thanks!

score 2 · Accepted Answer · edited Jul 09 '14 at 19:01

2

Well, it's not quite true. $\ker(g)$ is not a subset of $A$, so you can't take the quotient. The ring $C$ will be isomorphic to $A/\ker(g\circ f)$.

edited Jul 09 '14 at 19:01

Keenan Kidwell

26,020

answered Jul 09 '14 at 18:11

Sasha Patotski

7,161

score 1 · Answer 2 · answered Jul 09 '14 at 18:12

1

Well, it can't be since $\ker(g)$ is an ideal in $B$, but not in $A$.

Hint: What is the kernel of $g\circ f$?

answered Jul 09 '14 at 18:12

A.P.

9,728

Could the downvoter please explain her reasons? – A.P. Jul 09 '14 at 19:36

Composition of ring homomorphism

2 Answers2