EXTENDING MODULES WITH INJECTIVE OR SEMISIMPLE SUMMANDS

We characterize when the direct sum of an extending module and an inje ctive module is extending and when the direct sum of an extending modu le with the finite exchange property and a semisimple module is extend ing. We also characterize when the direct sum of a uniform-extending m odule and a semisimple module is uniform-extending and, in consequence , we prove that, for a right Noetherian ring R, an extending right R-m odule M-1 and a semisimple right R-module M-2, the right R-module M-1M-2 :is extending if and only if M-2 is M-1/Soc(M-1)-injective. (C) 19 98 Elsevier Science B.V. All rights reserved.