LSpecR. (b) The space LSpecR sober. Proof. 4 that the map P 1---+ (p : R) sends the left spectrum of the ring R into its Levitzki spectrum. 2) The map, which assigns to a subset V of SpeezR its radical - the intersection of all ideals from V - induces a bijection of the set of closed subsets of the space (SpeezR, T) onto the set of all the two- sided ideals a such that a = radle a).

This function satisfies the following compatibility condition: for any pair t : U ~ U', t' : U' ~ U" of inclusions, the diagram C. ',. (F(U")) Ft (1) F(U) is commutative. (r(U')) = ,(U) 0 Ft. The composition is defined by , o,'(U) = ,(U) o,'(U). We denote the category of presheaves in X by \l3te(X). We call a presheaf F quasi-coherent, or a structure presheaJ, if the morphism Ft is an isomorphism for any inclusion t. Qco~(X). One can see that the category of quasi-coherent presheaves is naturally equivalent to the category Cx.

And j 0 >" for some uniquely determined morphisms >. and >" respectively. ~ In+n' ~ n+n' Since 'P is an epimorphism and j is a monomorphism, there exists a unique R-module morphism h: n + n' ~ M such that ~ In+n'= j 0 h. (c) Finally, together with every ascending family W of ideals, the set C(e) contains the sum of all the ideals from W. (d) The assertions (b) and (c) allow to deduce (applying Zorn's Lemma) that the sum, nW, of all ideals from C(~) belongs to CW. Now, it follows from (a) that n(O belongs to F' 0 F'.