Noetherian Prime Rings Research Articles

On Noetherian prime rings

Classical left quotient rings are defined symmetrically. R is right (resp. left) quotient-simple in case R has a classical right (resp. left) quotient ring S which is isomorphic to a complete ring Dn of n X n matrices over a (not necessarily commutative) field D. R is quotient-simple if R is both left and right quotient-simple. Goldie [2] has determined that a ring R is right quotient-simple if and only if R is a prime ring satisfying the maximum conditions on complement and annihilator right ideals. In particular, any right noetherian prime ring is right quotient-simple. (See also Lesieur-Croisot [1 ] .) A (not necessarily commutative) integral domain K is a right Ore domain in case K possesses a classical right quotient field K. Observe that if K is a right Ore domain, then, for each natural number n, the ring Kn of all n X n matrices over K is right quotient-simple, and (K)n is its classical right quotient ring. A consequence of our main result (Theorem 2.3) is that the right quotientsimple rings can be determined as the class of intermediate rings of the extensions (K)n over Kn, n ranging over all natural numbers, and K ranging over all right Ore domains. Theorem 2.3 is much more precise. As a corollary we rederive a theorem of Goldie [3] on principal right ideal prime rings.