Why Does a Vector Space Have a Basis (Module Theory)

Module and vector spaceFirst we recall some backgrounds. Suppose $A$ is a ring with multiplicative identity $1_A$. A left module of $A$ is an additive abelian group $(M,+)$, together with an ring operation $A \times M \to M$ such that\begin{aligned}(a+b)x &= ax+bx \\a(x+y) &= ax+ay \\a(bx) &= (ab)x \\1_Ax &= x\end{aligned}for $x,y \in M$ and $a,b \in A$. As a corollary, we see $(0_A+0_A)x=0_Ax=0_Ax+0_Ax$, which shows $0_Ax=0_M$ for all $x \in M$. On the other hand, $a(x-x)=0_M$ which implies $a(...

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