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## Main Question or Discussion Point

I've recently encountered some forms of the second and third isomorphism theorem, but I don't quite get them. Could anyone explain in a bit of details please? I guess my thought was not in the right direction or something.

(Second isomorphism theorem) Let A be a subring and I an ideal of the ring R.

Show that there is an isomorphism of rings A/(A ∩ I) ∼= (A + I)/I

where A + I = {a + i : a ∈ A, i ∈ I}.

(Third isomorphism theorem) Let A be an ideal of the ring R. Show that if B is an

ideal of R that contains A then B/A is an ideal of R/A. Moreover the map B → B/A

is a bijection from the set of ideals of R containing A to the set of ideals of R/A. Show

that if B is an ideal of R that contains A then there is an isomorphism of rings

(R/A)/(B/A) ∼= R/B

Any help is greatly appreciated here!

(Second isomorphism theorem) Let A be a subring and I an ideal of the ring R.

Show that there is an isomorphism of rings A/(A ∩ I) ∼= (A + I)/I

where A + I = {a + i : a ∈ A, i ∈ I}.

(Third isomorphism theorem) Let A be an ideal of the ring R. Show that if B is an

ideal of R that contains A then B/A is an ideal of R/A. Moreover the map B → B/A

is a bijection from the set of ideals of R containing A to the set of ideals of R/A. Show

that if B is an ideal of R that contains A then there is an isomorphism of rings

(R/A)/(B/A) ∼= R/B

Any help is greatly appreciated here!