I am still confused about the difference between free module and finitely generated module. For example, $Z/2Z$ is finitely generated module, but why it is not finitely generated free module? What is difference between finitely generated module and finitely generated free module?I understand free module has a basis(not necessarily finite). In my opinion, Z/2Z can be viewed as a Z-module with basis 1+2Z, but why Z/2Z is not free module?


The "finitely generated" isn't really important here. You have to understand what are free modules.

A $R$-module $M$ is free if it has a basis, i.e., there exists $\{x_i\}_I \subset M$ (where $I$ can be chosen the be finite in the finitely generated case) such that the $x_i$'s form a $R$-basis of $M$.

This is a very special property, and every free module is isomorphic to $\bigoplus_I R$, for some $I$.

For your example, you have to be clear about which ring you are working with. $\Bbb{Z}/2$ is a free $\Bbb{Z}/2$-module with basis $\{1+2\Bbb{Z}\}$. However, it is not a $\Bbb{Z}$-module. Indeed, the only basis could be $\{1+2\Bbb{Z}\}$. But this is not a basis since $0=2\cdot (1+2\Bbb{Z})$.


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