Let be a ring with 1 and let be a left -module. Let be a nonempty chain of left -submodules of ; that is, if , then . Prove that is a submodule of .
We use the submodule criterion.
Certainly for some . Thus , and hence .
Now suppose and . Now and for some and . Suppose without loss of generality that ; then . Since is a submodule, . Thus .
Hence is a submodule of .