A stucture sheaf for semiprime rings

For any ring R (associative with 1) we associate a space X of prime torsion theories endowed with Golan's SBO-topology. A separated presheaf (L-,M) on X is then constructed for any right R-module MR, and a sufficient condition on M is given P such that L(-, M ) i s actually a sheaf. The sheaf space determined by L(-,M) represents M i n the following sense: M is isomorphic to the module of continuous global sections of p.These results are applied to the right R-module RR and it is seen that semiprime rings satisfy the required condition for L(-, R ) to be a sheaf. Among semiprime rings two classes are sin- gled out, fully symmetric semiprime and right noetherian semiprime rings; these two kinds of rings have the desirable property of yielding “nice” stalks for the above sheaf.