Let be a ring with 1, let be a left -module, and let be an element in the center of . Prove that is a submodule of . If is a field, , a left -module under multiplication, and , then is not a left -submodule of .

Let . We use the submodule criterion to show that is a submodule of . Note first that , so that is not empty. Now suppose and . Then . Thus is a submodule of .

Now for the example, note that . Letting and , we have . Thus is not a submodule of .