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If I remember well, the second chapter of

J. C. McConnell, J. C. Robson. Noncommutative Noetherian rings, vol. 30 of Graduate Studies in Mathematics (American Mathematical Society, Providence, RI, 2001)

contains a rather detailled proof of the Ore's theorem.

Edit: I just checked it on Google books and they allow zero divisors as well. The point is that if $S$ satisfies the Ore's condition (which is nothing but your first condition), then the set {$r\in R, rs=0$ for some $s\in S$} is an ideal in $R$ which is precisely the kernel of the natural map $R\rightarrow RS^{-1}$.

2 typo

If I remember well, the second chapter of

J. C. McConnell, J. C. Robson. Noncommutative Noetherian rings, vol. 30 of Graduate Studies in Mathematics (American Mathematical Society, Provi- denceProvidence, RI, 2001)

contains a rather detailled proof of the Ore's theorem.

1

If I remember well, the second chapter of

J. C. McConnell, J. C. Robson. Noncommutative Noetherian rings, vol. 30 of Graduate Studies in Mathematics (American Mathematical Society, Provi- dence, RI, 2001)

contains a rather detailled proof of the Ore's theorem.