Let’s start by recalling some background about modules.

Suppose that $R$ is a ring and $1_R$ is its multiplicative identity. A left $R$-module $M$ consists of an abelian group $(M, +)$ and an operation $R \times M \rightarrow M$ such that for all $r, s \in R$ and $x, y \in M$, we have:

1. $r \cdot (x+y)= r \cdot x + r \cdot y$ ($\cdot$ is left-distributive over $+$)
2. $(r +s) \cdot x= r \cdot x + s \cdot x$ ($\cdot$ is right-distributive over $+$)
3. $(rs) \cdot x= r \cdot (s \cdot x)$
4. $1_R \cdot x= x$

$+$ is the symbol for addition in both $R$ and $M$.
If $K$ is a field, $M$ is $K$-vector space. It is well known that a vector space $V$ is having a basis, i.e. a subset of linearly independent vectors that spans $V$.
Unlike for a vector space, a module doesn’t always have a basis. Continue reading A module without a basis →