Thread: Sum of Ideals

If A and B are both ideals, then A+B is an ideal. How can you show this?

Originally Posted by Ryaη

If A and B are both ideals, then A+B is an ideal. How can you show this?

What do elements from this ideal look like? Take an arbitrary element of this ideal and multiply it by an arbitrary ring element. Then apply the fact that multiplication in a ring is distributive.

Is it anything like this?

For some ring R that has A and B as ideals.

and , then . Thus, . Therefore .

Originally Posted by Ryaη

Is it anything like this?

For some ring R that has A and B as ideals.

and , then . Thus, . Therefore .

Not quite. Take . What does look like? Then take an arbitrary element from , call it . Then evaluate (and for completeness).

I think I got it. Thank you!