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  "body": "\n## Summary\n\nA noncommutative ring is a [[ring|0B1261B5ED]] in which the order of the factors in a product can matter.\nMatrix rings, quaternions, endomorphism rings and group rings of nonabelian groups are the main sources of examples.\nMany tools of commutative algebra split into left and right versions in this setting.\nThe study of these rings is the subject of noncommutative ring theory.\n\n## Explanations\n\n### L1 human\n\nIn ordinary arithmetic $ab = ba$, but many natural products do not behave this way.\nDoing one rotation of space and then another is in general not the same as doing them in the other order.\nSquare matrices model this: with\n$$A = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}, \\quad B = \\begin{pmatrix} 0 \u0026 0 \\\\ 1 \u0026 0 \\end{pmatrix},$$\none gets $AB = \\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 0 \\end{pmatrix}$ and $BA = \\begin{pmatrix} 0 \u0026 0 \\\\ 0 \u0026 1 \\end{pmatrix}$.\nSo the ring of $2 \\times 2$ real matrices is a noncommutative ring.\n",
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