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Math Help - Ideals in a ring R

  1. #1
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    Ideals in a ring R

    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? I think the best ring to represent R is the 2 by 2 matrix, since it's not commutative. But I already tried few particular matrix, but I couldn't find the one that satisfies the given condition. Appreciate someone's help. Thanks...
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  2. #2
    Lord of certain Rings
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    Your definition of I clearly shows that its a left ideal. For it not to be a right ideal, you would definitely want non-commutativity. So consider the set consisting of all n-by-n matrices whose last column is zero. The idea is that when you pre-multiply by any matrix(hence its a left Ideal), you will get a matrix that has last column 0. But:

    \begin{pmatrix} 1 & 0 \\ 2 & 0 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 2 & 2 \end{pmatrix} \notin I
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