Vol. 50, No. 2, 1974

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Orders with finite global dimension

Mark Bernard Ramras

Vol. 50 (1974), No. 2, 583–587

Let Λ be a lefl noetherian ring of finite left global dimension. Assume that Λ is quasi-local, i.e., that Λ modulo its Jacobson radical J(Λ) is a simple artin ring, and suppose that for some left Ore denominator set S contained in Λ,Σ = S1Λ is a left artin ring. Then Σ is a simple artin ring. More precisely, if Λ∕J(Λ)Mn(K), the ring of n by n matrices over a division ring K, then there exists an integer m dividing n such that Λ = Mn(Δ) and Σ = Mm(L), where L is a division ring and Δ is an order in L.

A corollary of this is that if the above Λ is local, then Σ is a division ring. Another corollary is that if F is a field and Λ is a left noetherian left order in Mp(F), and if Λ is quasi-locaI and l.gl. dim.Λ < , then Λ = Mp(R), where R F is the center of Λ.

Mathematical Subject Classification
Primary: 16A60
Received: 18 October 1972
Revised: 15 June 1973
Published: 1 February 1974
Mark Bernard Ramras