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A canonical arithmetic quotient for actions of lattices in by David Fisher

By David Fisher

During this paper we produce an invariant for any ergodic, finite entropy motion of a lattice in an easy Lie staff on a finite degree area. The invariant is largely an equivalence category of measurable quotients of a undeniable kind. The quotients are primarily double coset areas and are created from a Lie crew, a compact subgroup of the Lie staff, and a commensurability type of lattices within the Lie workforce.

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Conversely, assume a comaximal relation (6).

If P/JP is free over R/J , where J is the Jacobson radical of R, show that P is free over R. ) 12∗ . (Kaplansky [58]) Let P be a projective module over a local ring. Show that any element of P can be embedded in a free direct summand of P; deduce that every projective module over a local ring is free. 1 hold in most rings normally encountered, and counter-examples belong to the pathology of the subject. By contrast, the property defined below forms a significant restriction on the ring. Clearly any stably free module is finitely generated projective.

E. A = (0, I)P, B = P −1 (0, I)T . If we replace A, B by A P −1 , P B, we obtain C 0 0 A1 = 0 I A2 I B1 B2 0 . I On multiplying out, we find that A2 = 0, B2 = 0, C = A1 B1 and now the conclusion follows by induction. A square matrix C will be called a stable matrix atom if C ⊕ Ir is an atom for all r ≥ 1. 6. Over an Hermite ring every matrix atom is stable. We can specialize our ring still further. A ring R is called cancellable if for any projective modules P, Q, P ⊕ R ∼ = Q ⊕ R implies P ∼ = Q.

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