(1) Let M be an R-module and rR. Then prove that rM={rm:mM} and Mr={mM:rm=0} are submodules of M. (2) Prove that an R-module M is simple iff for every nonzero mM,M=Rm. (3) Prove that the R-module R is simple iff it is a division ring. (4) Let A,B,C be submodules of an R-module M and AB, A+C=B+C,AC=BC. Then prove that A=B. (5) An R-module M is said to be divect product of its submodules A&B iff M=A+B+AB={0}