# Thread: Ideals in a ring R

1. ## 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...

2. 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:

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