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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- Dec 17th 2008, 03:12 PManlysIdeals 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? - Dec 17th 2008, 03:26 PMByun
- Dec 17th 2008, 03:30 PManlys
- Dec 17th 2008, 04:17 PMNonCommAlg
$\displaystyle R=M_2(\mathbb{Z}), \ c=\begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix}.$ we have: $\displaystyle I=\{rc: \ r \in R \}=\left \{\begin{bmatrix}a & 0 \\ b & 0 \end{bmatrix}: \ a,b \in \mathbb{Z} \right \}.$ let $\displaystyle s=\begin{bmatrix}0 & 1 \\ 0 & 0 \end{bmatrix}.$ then $\displaystyle cs=s \notin I. \ \Box$