Let be a ring with 1 and let be a left -module. Suppose is generated by elements, say . Show that if is a submodule then has a generating set containing at most elements. Conclude that every quotient of a cyclic module is cyclic.
We claim that . The direction is clear. To see the direction, suppose with . Then as desired. Thus is an -module generating set for . (Which may contain redundant elements.)
In particular, If is cyclic, then it has a singleton generating set. Every quotient of is then generated by at most one element; since every module is generated by at least one element, every quotient of is also cyclic.