Let be a ring with 1 and let be a left -module. Let be a submodule. We define the annihilator of in to be . Prove that is a two-sided ideal in .
We need to show that is nonempty, closed under subtraction, and absorbs from both sides.
Let ; certainly . Thus .
Let and . Then , so that .
Let , let and let . Note that and , so that .
Thus is a two-sided ideal in .