MathDB

2. Show that a ring

R

has a unit element if and only if any

R

-module

G

can be written as a direct sum of

RG

and of the trivial submodule of

G

. (An

R

-module is a linear space with

R

as its scalar domain.

RG

denotes the submodule generated by the elements of the form

rg

(

r \in R, g \in G

). The trivial submodule of

G

consists of the elements

g

G

for which

rg=0

holds for every

r \in R

.) (A. 20)

Problem Statement