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2) and the equality of the entropies of Xx and X 2 . RESOLVING MAPS AND THE DIMENSION GROUP 47 Now we show that if 7r(x) = 7r(x'), then

2 B , om )-*(Y, a m ). e. maps ( X ^ , o)-»(Xit a) lift to closing maps between the canonical covers. • The condition (2) is given some teeth in section 10. 5. LINEAR MAPS, IDEAL CLASSES AND REALIZATION For convenient reference, we are assembling some basic algebraic results in this section. First, we give several conditions for an abstract quotient of a dimension group pair to be a dimension group pair in its own right. Then we consider certain ideal classes, fundamental to the results of sections 6 and 7.

R = u(A k r) —= (uA k )r — e[r] . Next, we show that 0 is a Z[l/A]-ideal. To see this, first note that J is an additive subgroup since it is the image of G A via a linear map. Also, ^<=z[l/A] since «^ c [r]. Moreover, J is (l/A)-in variant since for any ueG A , MA(U) = MA OA(A~1(U)) = AMA(A~1(U)) by (1), hence -i- M A (u) = M A ( A ~ 1 ( u ) ) 6 ^ . So, & is an additive subgroup of Z[l/A] closed under multiplication by 1/A, hence J is aZ[l/A]-ideal. RESOLVING MAPS AND THE DIMENSION GROUP 55 Now, we have that & is a Z[l/A]-ideal contained in [r].

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