Span and Smallest Submodule Proof

Question

Let R be a ring, M a R-module, and $X \subseteq M$

Show that span$(X)$ is the smallest submodule of R containing X.

My ideas: Every submodule is contained in its span so $X \subseteq$ span$(X)$ and $x \in X \subseteq$ span(X) but now I need to show that it's the smallest... how do I go about showing this?

Any advice would be great!

Answer 1

Let $J$ be any submodule containing $X$. Then show that $J$ contains span$(X)$.

This is not very difficult: use the definition of span($X$) to show that since $J$ is a module containg $X$ it should contain span$(X)$.