Left ideal

Remarks

A right ideal is defined in the same way with $ar \in I$ in clause 2. A two-sided ideal is a subset that is both a left and a right ideal. Example: let $K$ be a field and let $I \subseteq M_2(K)$ be the set of matrices whose second column is zero. It is an additive subgroup, and for $A \in M_2(K)$ and $X \in I$ the second column of $AX$ is $A$ times the second column of $X$, which is zero. So $I$ is a left ideal. Non-example: the set $J$ of matrices whose second row is zero is an additive subgroup, but $E_{21} E_{11} = E_{21} \notin J$, where $E_{ij}$ is the matrix with $1$ in position $(i,j)$ and $0$ elsewhere. So clause 2 fails for $J$; it is a right ideal instead. The left ideals of $R$ are the submodules of $R$ as a left module over itself. For each $a \in R$, the set $Ra = \set{ra : r \in R}$ is the left ideal generated by $a$.