# Ideals and homomorphisms

• May 18th 2009, 09:40 PM
curiousmuch
Ideals and homomorphisms
Let I and J be ideals of a ring R and let I be contained in J.

Show that J/I is an ideal of R/I and also show that (R/I)/(J/I) is isomorphic to R/J.

For the second part I am just trying to use a ring homomorphism but am having trouble defining a map.

Also if there is a better way to do the first part than to show that J/I is a sub ring and then an ideal that would be awesome.

Thanks.
• May 18th 2009, 09:47 PM
NonCommAlg
Quote:

Originally Posted by curiousmuch
Let I and J be ideals of a ring R and let I be contained in J.

Show that J/I is an ideal of R/I and also show that (R/I)/(J/I) is isomorphic to R/J.

For the second part I am just trying to use a ring homomorphism but am having trouble defining a map.

Also if there is a better way to do the first part than to show that J/I is a sub ring and then an ideal that would be awesome.

Thanks.

for the first part just do what we usually do to prove something is an ideal. for the second part define $f: R/I \longrightarrow R/J$ by $f(r+I)=r+J.$ this map is well-defined because $I \subseteq J.$

it's obviously a surjective ring homomorphism and $\ker f = J/I$ and the result follows.