Orders with finite global dimension

Mark Bernard Ramras

Vol. 50 (1974), No. 2, 583–587

DOI: 10.2140/pjm.1974.50.583

Abstract

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 = S⁻¹Λ is a left artin ring. Then Σ is a simple artin ring. More precisely, if Λ∕J(Λ)≅M_n(K), the ring of n by n matrices over a division ring K, then there exists an integer m dividing n such that Λ = M_n(Δ) and Σ = M_m(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 M_p(F), and if Λ is quasi-locaI and l.gl. dim.Λ < ∞, then Λ = M_p(R), where R ⊂ F is the center of Λ.

Mathematical Subject Classification

Primary: 16A60

Milestones

Received: 18 October 1972

Revised: 15 June 1973

Published: 1 February 1974