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Math Help - Sum of Ideals

  1. #1
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    Sum of Ideals

    If A and B are both ideals, then A+B is an ideal. How can you show this?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Ryaη View Post
    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.
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  3. #3
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    Is it anything like this?

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

    nA \in R and nB \in R, then nA+nB \in R. Thus, n(A+B) \in R. Therefore (A+B) \in R.
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Ryaη View Post
    Is it anything like this?

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

    nA \in R and nB \in R, then nA+nB \in R. Thus, n(A+B) \in R. Therefore (A+B) \in R.
    Not quite. Take x \in A + B. What does x look like? Then take an arbitrary element from R, call it r. Then evaluate xr (and rx for completeness).
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  5. #5
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    I think I got it. Thank you!
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