GOLDIE-DIMENSION OF A SUM OF MODULES

The formula dim(A+B)=dim(A)+dim(B)-dim(A and B) works when 'dim' stand s for the dimension of subspaces A,B f any vector space. In general, h owever, it does no longer hold if 'dim' means the uniform (or Goldie) dimension of submodules A,B of a module M over a ring R, and in fact t he left hand side may be infinite while the right had side is finite. In this paper we shall give a characterization of those modules M in w hich the formula holds for any two submodules A,B, as well as some con ditions In the ring R which guarantee that dim(A+B) is finite whenever A and B are finite dimensional R-modules.