What's a Module / Submodule?

Before we introduce what is a module let's recall some definitions. A ring is a set $R$ equipped with addition which is associative, commutative, with identity and inverse (with only these properties we call it an Abelian group), multiplication which is associative, with identity, and distributive with respect to addition. If multiplication is commutative we call it a commutative ring.

Now,

Definition 1. A left $R$-module for a commutative ring $R$ is an additive Abelian group $M=(M,+)$ with a left multiplication by elements of $R$, and this multiplication satisfies:

1.("associative") $(r_1\cdot_R r_2)m=r_1\cdot_M (r_2\cdot_M m)$

2.("distributive") $(r_1+_R r_2)\cdot_M m=r_1\cdot_M m+_M r_2\cdot_M m, \ \ r\cdot_M (m_1+_M m_2)=r\cdot_M m_1+_M r\cdot_M m_2$

3.(identity) $1_r\cdot m=m$

for any $r_1, r_2\in R$ and $m\in M$, where $\cdot_R$ means multiplication in $R$ and so on. Right module can be defined in similar way.

Some basic verifications!

An somewhat amazing fact you can quickly see: when $R$ is a field, a $R$-module becomes a vector space! We call $M$ in this case a $R$-vector space.

Notice that since modules can be viewed as generalization of vector spaces, things like direct sum, linear independent, basis etc. also well-defined here. We can even extend nice results like dimension theorem:

Theorem 2. Any two basis of a $R$-module has the same size.

Submodules, however, is analogous to subspace too:

Definition 3. Let $M$ be a left $R$-module. A submodule $N$ is a left $R$-module whose elements form a subset of $M$'s.

Finally, some more definitions about modules:

Definition 4. A $R$-module is called free if it has a basis, and we call the size of this basis rank.

Still, not much different from vector spaces.