# Thread: Herstein: Subrings and Ideals

1. ## Herstein: Subrings and Ideals

Herstein's Abstract Algebra: Ch 4. Sect 3. # 5.

If I is an ideal of R and A is a subring of R, show that I ∩ A is an ideal of A

2. ## Re: Herstein: Subrings and Ideals

Originally Posted by ThatPinkSock
Herstein's Abstract Algebra: Ch 4. Sect 3. # 5.

If I is an ideal of R and A is a subring of R, show that I ∩ A is an ideal of A
So where exactly are you stuck?

To show a set is an ideal you need to varify two things.

1. The set is a sub group under "+"

2. That the set "absorbs" elements under multipication.
For any element r in the ring and any elment x in the ideal that
xr is in the ideal and rx is in the ideal.

Try to show the above and post back if you get stuck.

3. ## Re: Herstein: Subrings and Ideals

(1) [ I ∩ A, +] is a group because [I, +] and [A, +] are subgroups of R and the intersection of two subgroups is a subgroup.

(2) Let x ∈ I ∩ A and let r ∈ A. Then x ∈ I and x ∈ A. Since I is an ideal of R, xr ∈ I and rx ∈ I. Since x ∈ A, xr ∈ A and rx ∈ A. So xr ∈ I ∩ A and rx ∈ I ∩ A.

From (1) and (2) I ∩ A is an ideal of A.