Second and third isomorphism theorem

In summary, the second isomorphism theorem states that there is an isomorphism between the quotient ring A/(A ∩ I) and the quotient ring (A + I)/I, where A is a subring and I is an ideal of the ring R. The third isomorphism theorem states that if A is an ideal of R and B is an ideal of R that contains A, then B/A is an ideal of R/A. Furthermore, the map B → B/A is a bijection between the set of ideals of R containing A and the set of ideals of R/A. Additionally, there is an isomorphism between (R/A)/(B/A) and R/B. These theorems are often encountered in
  • #1
rainwyz0706
36
0
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!
 
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  • #2
Where does the theorem stop making sense? Also what book are you using and have you encountered these theorems in group theory?
 

Related to Second and third isomorphism theorem

1. What is the second isomorphism theorem?

The second isomorphism theorem, also known as the diamond isomorphism theorem, states that if H and K are subgroups of a group G, where H is normal in G, then the quotient group HK/K is isomorphic to H/(H ∩ K).

2. How is the second isomorphism theorem used in group theory?

The second isomorphism theorem is used to simplify the study of quotient groups and their subgroups. It allows for the comparison of different quotient groups and their corresponding subgroups, making it a powerful tool in group theory.

3. What is the third isomorphism theorem?

The third isomorphism theorem, also known as the correspondence theorem, states that if N and H are normal subgroups of a group G, where N is a subgroup of H, then the quotient group H/N is isomorphic to (G/N)/(H/N).

4. How does the third isomorphism theorem relate to the first and second isomorphism theorems?

The third isomorphism theorem is a generalization of the first and second isomorphism theorems. It allows for the comparison of quotient groups with multiple normal subgroups, whereas the first and second theorems only consider one normal subgroup.

5. What is the significance of the second and third isomorphism theorems?

The second and third isomorphism theorems are important tools in group theory that allow for the simplification and comparison of quotient groups and their subgroups. They also provide a deeper understanding of the structure of groups and their relationships with their subgroups.

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