Show that an ideal is proper if and only if it does not contain 1.
Prove these two statements: If 1 is in the ideal, it is not proper. If 1 is not in the ideal, it is proper. The second statement is obvious by the definition of being a proper ideal, i.e. if the ring is and ideal is , since .
Use the definition of an (left) ideal: for all elements of the ring, . In particular, if then for any . What if ?