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Any ring $A$ has the canonical structure of an $A$-module via left multiplication. The submodules of this module are precisely the left ideals of $A$, by definition of left ideal.

Modules of the form$$\bigoplus_{i \in I} A,$$a direct sum of copies of the module $A$ labeled by a (possibly infinite) set $I$, are called free modules. We say that $A^n$ is a free module of rank $n$.

My question is, why would we care about free modules?

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    $\begingroup$ For instance because every module can be looked at as quotient-module of a free module. There are more reasons. If you want to build a module starting with some set, and do that efficiently as possible then you end up with a free module. $\endgroup$
    – drhab
    Commented Sep 21, 2015 at 14:01
  • $\begingroup$ In addition to @rschwieb answer they have universal property. Perhaps they have mentioned it indirectly in the first point. $\endgroup$
    – Math137
    Commented Sep 21, 2015 at 16:05

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  1. They are perhaps the easiest type of module to understand. They are the most vector space-like modules, in a sense. For example, the endomorphism rings of finitely generated free $R$ modules are just square matrix rings over $R$.

  2. They contain enough information to recover all Morita equivalent rings. To do that, you just need to know all the finitely generated projective modules, and those are finitely generated direct summands of (you guessed it) free modules.

  3. They are homologically special. If you believe that projective and injective modules are special, then just think that free modules are very special projective modules. You can even use free resolutions to analyze some modules.

  4. Every module is a quotient of a free module. (Already mentioned by others in the comments.

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