Geodesic
A Shorter Proof of Goldie's Theorem
Canadian mathematical bulletin, Tome 12 (1969) no. 5, pp. 597-602

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In this note we present an extremely short proof of Goldie's theorem on the structure of semiprime Noetherian rings [1]. The outline of the proof was given by Procesi and Small in [4]. By utilizing the concept of the singular ideal of a ring we have been able to weaken the hypotheses of many of the steps in [4]. Most significantly, we are able to avoid a reduction to the case of prime rings, and in Lemma 5 we give an informative list of the relationship between regular elements and essential ideals of semiprime rings. 
Zelmanowitz, Julius. A Shorter Proof of Goldie's Theorem. Canadian mathematical bulletin, Tome 12 (1969) no. 5, pp. 597-602. doi: 10.4153/CMB-1969-077-1
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     author = {Zelmanowitz, Julius},
     title = {A {Shorter} {Proof} of {Goldie's} {Theorem}},
     journal = {Canadian mathematical bulletin},
     pages = {597--602},
     year = {1969},
     volume = {12},
     number = {5},
     doi = {10.4153/CMB-1969-077-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1969-077-1/}
}
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