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  "body": "\n## Summary\n\nA left ideal of a [[ring|0B1261B5ED]] $R$ is an additive subgroup that is closed under multiplication on the left by every element of $R$.\nLeft ideals are exactly the submodules of $R$ viewed as a left module over itself.\nIn a commutative ring left ideals, right ideals and [[two-sided ideals|0CD87SFM2N]] all coincide with [[ideals|096Z9MP2ME]].\nIn a noncommutative ring the three notions differ, and left ideals carry much of the module theory of $R$.\n\n## Explanations\n\n### L1 human\n\nA left ideal is a part of a ring that absorbs multiplication from the left: multiply one of its elements on the left by anything, and the product stays inside.\nMultiplication on the right may lead out of it.\nIn the ring of $2 \\times 2$ matrices, the matrices whose second column is zero form a left ideal.\nMultiplying such a matrix on the left by any matrix keeps the second column zero.\nMultiplying it on the right can fill the second column, so it is not a right ideal.\n",
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