= Goldie's theorem
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{title2=$R\text{ semiprime right Goldie}\iff Q_{\mathrm{cl}}^r(R)\text{ semisimple Artinian}$}
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= Goldie theorem
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A <semiprime ring> has a <classical right ring of quotients> that is a <semisimple ring> and <Artinian> exactly when it is a <right Goldie ring>. In particular, the theorem applies to a <semiprime ring> that is a <right Noetherian ring>. It extends the fraction-field construction from commutative domains to <rings> whose quotient can be a finite product of <matrix> <rings> over division <rings>.
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