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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nA *left ideal* of $R$ is a subset $I \\subseteq R$ such that:\n\n1. $I$ is a subgroup of the additive [[group|G5S1M9]] $(R, +)$.\n2. $ra \\in I$ for all $r \\in R$ and $a \\in I$.\n\n## Remarks\n\nA *right ideal* is defined in the same way with $ar \\in I$ in clause 2.\nA [[two-sided ideal|0DDA6CY7GJ]] is a subset that is both a left and a right ideal.\nExample: let $K$ be a field and let $I \\subseteq M_2(K)$ be the set of matrices whose second column is zero.\nIt 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.\nSo $I$ is a left ideal.\nNon-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.\nSo clause 2 fails for $J$; it is a right ideal instead.\nThe left ideals of $R$ are the submodules of $R$ as a left module over itself.\nFor each $a \\in R$, the set $Ra = \\set{ra : r \\in R}$ is the left ideal generated by $a$.\n",
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