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  "body": "\n## Statement\n\nLet $R$ be a [[prime ring|Z9TJSZZYG1]] that satisfies a monic [[polynomial identity|C0EDT81EGH]].\nLet $Z$ be the [[center|MVT64B3B6H]] of $R$, and let $S = Z \\setminus \\set{0}$.\n\n1. $Z$ is a commutative domain, and each element of $S$ is a regular element of $R$.\n2. The ring $Q = R S^{-1}$, obtained by inverting the nonzero central elements, is a finite-dimensional [[central simple algebra|5HRKJBAY2V]] over the [[field of fractions|0F2PTGX1A2]] $F$ of $Z$.\n3. $Q$ is the [[classical ring of quotients|C5971PGKPH]] of $R$ on both sides.\n\n## Remarks\n\nPosner proved that a prime PI ring has a classical ring of quotients that is a simple Artinian ring finite-dimensional over its centre.\nThe sharper statement that one only has to invert the nonzero central elements, given in parts 1 and 2, is a later refinement based on [[central polynomials|667YZTHE96]].\nOne way to see the original statement: a prime PI ring satisfies the conditions of a [[Goldie ring|QFDC3NYHJE]], so [[Goldie's theorem|MDS8655N0Z]] gives a simple Artinian quotient ring $Q \\cong M_s(D)$.\nThe ring $Q$ satisfies the same multilinear identities as $R$, and then [[Kaplansky's theorem|8NE9Q44FEG]] shows that $Q$ is finite-dimensional over its centre.\n\nExample: $R = M_n(\\Z)$ is prime and satisfies $s_{2n}$, its centre is $\\Z$, and $Q = M_n(\\Q)$, a central simple algebra over $\\Q$.\n\nThe primeness hypothesis is needed for the conclusion that $Q$ is simple.\nThe ring $\\Z \\times \\Z$ is commutative, hence PI, but it is not prime, and its classical ring of quotients $\\Q \\times \\Q$ is not simple.\n\n## Explanations\n\n### L1 human\n\nA commutative domain sits inside its field of fractions.\nPosner's theorem says that a prime ring satisfying a polynomial identity sits, in the same way, inside a matrix-like algebra over a field: a central simple algebra of finite dimension.\nOne gets there just by dividing by central elements, as for fractions of integers.\n\n### L2 human\n\nPosner's theorem reduces the study of prime PI rings to orders in central simple algebras.\nThe dimension of $Q$ over $F$ is a square $n^2$, and $n$ is called the PI degree of $R$.\nCombined with the theory of [[generic matrix algebras|GJGEVMZ828]], it is the main tool for questions about prime PI rings, such as their dimension theory and their centres.\n",
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