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?
$\displaystyle R=M_2(\mathbb{Z}), \ c=\begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix}.$ 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 $\displaystyle cs=s \notin I. \ \Box$