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  "body": "\n## Statement\n\nLet $R$ be a ring with identity.\n\n1. $R$ is [[semiprime|5F0YMCV50R]] and right [[Goldie|QFDC3NYHJE]] if and only if $R$ has a [[semisimple|NY25HXA478]] right [[classical ring of quotients|C5971PGKPH]].\n2. $R$ is [[prime|Z9TJSZZYG1]] and right Goldie if and only if $R$ has a [[simple|62M0NC3E2T]] [[Artinian|20QEAM8AX2]] right classical ring of quotients.\n\n## Remarks\n\nFor the forward direction, one shows that in a semiprime right Goldie ring a right ideal is [[essential|8MS51FASMX]] exactly when it contains a regular element.\nFrom this one gets that the regular elements satisfy the right [[Ore condition|TJP4ARC2X7]], so $Q$ exists.\nOne also gets that $Q$ has no proper essential right ideals, which forces $Q$ to be semisimple.\nFor the converse, the Goldie conditions and semiprimeness pass from the semisimple ring $Q$ down to its right order $R$.\n\nExample: a commutative integral domain $R$ is prime and Goldie, and its classical ring of quotients is its field of fractions, a simple Artinian ring.\nExample: every semiprime right [[Noetherian ring|0670T59HGE]] has a semisimple right classical ring of quotients, since right Noetherian rings are right Goldie.\nSo a Weyl algebra over a field, which is a Noetherian domain, has a division ring of quotients.\n\nThe semiprime hypothesis is needed.\nThe ring $\\Z/4\\Z$ is Noetherian, hence Goldie, but it is not semiprime, because the ideal $2\\Z/4\\Z$ has square $0$.\nIts regular elements are $1$ and $3$, which are already units, so its classical ring of quotients is $\\Z/4\\Z$ itself, which is not semisimple.\n\n## Explanations\n\n### L1 human\n\nEvery integral domain sits inside a field of fractions, and this is how one builds $\\Q$ from $\\Z$.\nGoldie's theorem is the noncommutative version.\nA ring with no nilpotent ideals and with mild finiteness conditions sits inside a ring of fractions, and this ring of fractions is as nice as possible.\nBy the [[Artin–Wedderburn theorem|R1F271180N]], it is a finite product of matrix rings over division rings.\n\n### L2 human\n\nGoldie's theorem is the starting point of the theory of noncommutative Noetherian rings.\nIt lets one study a prime Noetherian ring $R$ through its quotient ring $Q \\cong M_n(D)$, with $D$ a division ring, and define invariants such as the Goldie rank $n$.\nIt applies to enveloping algebras of finite-dimensional Lie algebras over a field and to Weyl algebras, which are Noetherian domains.\n",
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