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  "body": "\n## Summary\n\nA polynomial identity ring, or PI ring, is a [[ring|0B1261B5ED]] in which some nonzero noncommutative polynomial vanishes for all choices of elements.\nCommutative rings are PI rings, because $xy - yx$ vanishes on them, and so are matrix rings over commutative rings by the [[Amitsur–Levitzki theorem|YDH60KPTWT]].\nPI rings form the class of noncommutative rings that are closest to commutative rings, and many results of commutative algebra extend to them.\nTheir theory is built on [[free algebras|2P3Y2FRCSZ]], [[central polynomials|667YZTHE96]] and [[generic matrices|GJGEVMZ828]].\n\n## Explanations\n\n### L1 human\n\nIn a commutative ring the rule $xy = yx$ holds for all elements.\nA PI ring is a ring where some other, more complicated rule of this kind holds for all elements.\nFor $2 \\times 2$ matrices, $xy$ and $yx$ differ, but the rule $(xy - yx)^2 z = z (xy - yx)^2$ always holds.\nSuch a rule is called a polynomial identity, and its presence forces the ring to behave much like a commutative ring or a ring of matrices.\n",
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