Every quotient of a finitely generated module is finitely generated

Let R be a ring with 1 and let M be a left R-module. Suppose M is generated by n elements, say \{a_i\}_{i=1}^n. Show that if N \subseteq M is a submodule then M/N has a generating set containing at most n elements. Conclude that every quotient of a cyclic module is cyclic.

We claim that M/N = (a_i + N \ |\ i \in [1,n]). The (\supseteq) direction is clear. To see the (\subseteq) direction, suppose x + N \in M/N with x = \sum r_i \cdot a_i. Then x + n = \sum r_i \cdot (a_i + N) as desired. Thus \{a_i + N\}_{i=1}^n is an R-module generating set for M/N. (Which may contain redundant elements.)

In particular, If M is cyclic, then it has a singleton generating set. Every quotient of M is then generated by at most one element; since every module is generated by at least one element, every quotient of M is also cyclic.

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