Modules
Let \(R\) be a ring. An \(R\)module or module over \(R\) is an abelian group \((M, +)\) and a map \(\mu : R\times M \rightarrow M\) (scalar mulitplication) such that for all \(a,b \in R\) and \(x,y \in M\) we have

\(a(x+y) = a x + a y\)

\((a+b)x = a x + b x\)

\((a b)x = a (b x)\)

\(1 x = x\)
Alternatively, we may say that an \(R\)module \(M\) is an abelian group with a ring homomophism \(R \rightarrow End(M)\) where \(End(M)\) is the ring of endomorphisms of \(M\).
Example

Any ideal \(I\triangleleft R\) is an \(R\)module. (Scalar multiplication is ring multiplication.) In particular \(R\) is an \(R\)module
2. If \(R\) is a field then \(R\)modules are precisely the vector spaces over \(R\).
3. All abelian groups are \(\mathbb{Z}\)modules.
4. Let \(R = K[x]\) for some field \(K\). Then an \(R\)module is a \(K\)vector space together with some linear transformation.
5. Let \(K\) be a field, \(G\) be a group and consider the group ring \(R = K[G]\). Then \(R\) modules are precisely \(K\)representations of \(G\).
A mapping \(f:M\rightarrow N\) between \(R\)modules \(M,N\) is an \(R\)module homomorphism or \(R\)linear if addition and scalar multiplication are preserved, that is for all \(x,y \in M\) and \(a \in R\) we have \(f(x+y) = f(x) + f(y), f(a x ) = a f(x)\). Alternatively we may say \(f\) is a homomorphism between abelian groups that respects the actions of the ring.
Let \(Hom_R (M,N)\) be the set of all \(R\)module homomorphisms from \(M\) to \(N\). For all \(f,g \in Hom(M,N), a \in R\), define \(f+g\) and \(a f\) by \((f + g)(x) = f(x) + g(x), (a f) (x) = a f(x)\). Then it can be easily verified that \(Hom(M,N)\) is an \(R\)module.
Let \(u:M'\rightarrow M\) and \(v:N\rightarrow N'\) be \(R\)module homomorphisms. They induce \(R\)module homomorphisms \(\bar(u) = f u : Hom(M,N)\rightarrow Hom(M',N)\) and \(\bar(v) = v f : Hom(M,N) \rightarrow Hom(M,N')\).
Example: If \(R\) is a field then \(R\)module homomorphisms are linear transformations which may be written as matrices. The induced homomorphisms can be computed via matrix multiplications.
For any \(R\)module \(M\) we have \(Hom_R(R,M) \cong M\). This can be seen by considering the map given by \(f \mapsto f(1)\) for all \(f \in Hom_R(R,M)\).