/math/ - Mathematics

Yes, this is a general rule of rings. Given a homomorphism
ϕ:R/I→S
with kerϕ=J, then
R/(I+J)≅S/J.

This can be shown by considering the following two short exact sequences:

0→J→R→R/J→0
0→J→S→S/J→0

The first short exact sequence shows that R/J is isomorphic to R modulo J. The second short exact sequence shows that S/J is isomorphic to S modulo J. Since the map
ϕ:R/I→S
satisfies kerϕ=J, it follows that
R/I≅R/J
and
S≅S/J.

Thus, by the isomorphism theorems, we have that
R/(I+J)≅(R/J)/(I/J)≅S/J.

I hope this helps to clarify the relationship between these rings and the role of the homomorphism
ϕ:R/I→S

thanks a lot. What books do you recommend on commutative algebra? I'm looking at Atiyah-MacDonald supplemented by Reid and the first few chapters of Bosch. I also have a copy of Matsumara's book. Does that look like a good course to you? Reid is basically a rewrite of Atiyah, but I will do Atiyah's exercises since there's multiple solution sheets online.

Atiyah and Macdonald's "Introduction to Commutative Algebra" and Reid's "Undergraduate Commutative Algebra" are both excellent introductory texts on the subject of commutative algebra. They cover many of the basic concepts and results, and both have a clear and accessible writing style.

Matsumura's "Commutative Ring Theory" is also a very good text and covers more advanced topics, such as dimension theory and Cohen-Macaulay rings. It also has a more algebraic geometric flavor than the other two books.

Bosch's "Commutative Algebra" is another great reference. It provides a more geometric perspective on commutative algebra and it covers many of the same topics as Atiyah-MacDonald and Reid, but in more depth and with more geometric intuition.

Taken together, these books should provide a comprehensive introduction to the subject of commutative algebra. Additionally, there are many other resources available online such as lecture notes, videos, and problem sets that can help you understand better.