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Math Help - Ideals in a ring

  1. #1
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    Ideals in a ring

    Let c be in a ring R and let I = {rc | r is in R}
    Give an example to show that if R is not a commutative ring, then I need not be an ideal.

    Can someone help me this?
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  2. #2
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    Quote Originally Posted by anlys View Post
    Let c be in a ring R and let I = {rc | r is in R}
    Give an example to show that if R is not a commutative ring, then I need not be an ideal.
    For any k \in R and any rc \in I, k(rc) \in I.
    But (rc)k may not be in R.
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  3. #3
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    Quote Originally Posted by Byun View Post
    For any k \in R and any rc \in I, k(rc) \in I.
    But (rc)k may not be in R.
    Hello Byun,
    Thanks for your input. Do you also have a particular example to show this?
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  4. #4
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    Quote Originally Posted by anlys View Post
    Let c be in a ring R and let I = {rc | r is in R}
    Give an example to show that if R is not a commutative ring, then I need not be an ideal.

    Can someone help me this?
    R=M_2(\mathbb{Z}), \ c=\begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix}. we have: I=\{rc: \ r \in R \}=\left \{\begin{bmatrix}a & 0 \\ b & 0 \end{bmatrix}: \ a,b \in \mathbb{Z} \right \}. let s=\begin{bmatrix}0 & 1 \\ 0 & 0 \end{bmatrix}. then cs=s \notin I. \ \Box
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