example of right / left ideals

Question

I searched and searched for examples of right / left ideals, but could find none. I read that a right ideal of $S$ is a subset of $R$ of $S$ such that $RS \subseteq R$, and that symetrically, a left ideal is a subset $L$ of $S$ such that $SL \subseteq L$. What concrete examples of right / left ideals are there?

Answer 1

I don't know where did you look, but one can find many examples of right/left ideals in any standard textbook. The fact that we can find such ideals in any ring $S$ is easy to see but to have them different, we need a non-commutative ring in which multiplication from left and right makes a difference. For example,

1) If our ring is commutative, like $\mathbb{Z}$, we have that any ideal, say $n\mathbb{Z} $ is both a right/left ideal. To see this,

$ \because (n\mathbb{Z})(\mathbb{Z}) = (\mathbb{Z})(n\mathbb{Z}) \subseteq n\mathbb{Z}. $

2) Then a difference occurs only when we have a non-commutative ring. A first example that occurs to one's mind is that of matrices. For instance, consider all $3\times 3$ matrices over $\mathbb{R}$. Then consider the following ideals :

$R_1 = \left\{ \begin{bmatrix} a & b & c \\\ 0 & 0 & 0 \\\ 0 & 0 & 0\end{bmatrix} : a,b,c \in \mathbb{R}\right\}$ is a right ideal of $M_{3}(\mathbb{R})$ and $ R_2 = \left\{ \begin{bmatrix} a & 0 & 0 \\\ b & 0 & 0 \\\ c & 0 & 0\end{bmatrix} : a,b,c \in \mathbb{R}\right\}$ is a left ideal of $M_{3}(\mathbb{R}).\ $ Here we can see that $R_1 \ne R_2$.

Hope this example helps, you can find more by searching for them on the internet/any book .