Module and Ring Is module a special case of a ring?
Someone told me, "No, 'module a ring' is similar to 'vector space over a field.'"
If his idea is true, could you explain it a little bit?
 A: In a ring $R$ you can multiply any two elements in the ring, and also add them. For instance $\frac{\mathbb{Z}[x]}{(x^2+1)}$ is a ring since you can multiply any two elements (and add them) and they are still in the ring.
The idea with a module is that you are not allowed to multiply everything you want to multiply, but only certain things. (This is just like vector spaces, you can not multiply two vectors, but you can multiply a vector by a scalar). How would this look like with $\frac{\mathbb{Z}[x]}{(x^2+1)}$? First of all you have to specify: what am I allowed to multiply? If any element, such as $x+3\in\frac{\mathbb{Z}[x]}{(x^2+1)}$, is allowed to be multiplied by an integer (inside $\mathbb{Z}$) this is now called a $\mathbb{Z}$-module. This is like a vector space (but not quite) because the elements $ax+b$ can be thought of as vectors, and you are allowed to multiply them by integers (scalars).
If we replace $\mathbb{Z}$ by $\mathbb{R}$ (or any other field) then $\frac{\mathbb{R}[x]}{(x^2+1)}$ is by definition a vector space.
A: A module over a ring $R$ is a “representation of $R$”. More precisely, let $M$ be a left module over $R$; then we get a ring homomorphism
$$
\lambda_M\colon R\to\operatorname{End}(M)
$$
where $\operatorname{End}(M)$ denotes the ring of homomorphisms of $M$ regarded as an abelian group. The homomorphism $\lambda_M$ associates to $r\in R$ the map $\lambda_M(r)=\hat{r}$ where $\hat{r}(x)=rx$.
The module axiom $r(x+y)=rx+ry$ tells us that $\hat{r}\in\operatorname{End}(M)$. The axiom $(r+s)x=rx+sx$ tells us that $\lambda_M$ is a homomorphism with respect to addition; the axiom $r(sx)=(rs)x$ tells us that $\lambda_M$ is a homomorphism with respect to multiplication (which is map composition in $\operatorname{End}(M)$). The module axiom $1x=x$ yields that $\lambda_M$ preserves the identity.
Conversely, if $M$ is an additive abelian group, a ring homomorphism $\varphi\colon R\to\operatorname{End}(M)$ defines a left module structure on $M$ by
$$
rx=\varphi(r)(x)
$$
as can be readily verified.
The idea behind modules is that we get to know (some of) the properties of $R$ by looking at its representation over abelian groups. Not precisely all properties, because of Morita theorems: modules over a ring only characterize the ring up to a certain extent. For instance, $R$ and the matrix ring $M_n(R)$ have essentially the same representations (in other terms, they have equivalent categories of modules).
Another way to see modules is as generalizations of vector spaces, where the base field is changed into a general ring. An example of this is the classical technique of studying a particular endomorphism $f$ of a finite dimensional vector space $V$ over the field $K$ by making $V$ into a module over the polynomial ring $K[X]$, where the action of $a_0+a_1X+\dots+a_nX^n$ over $v\in V$ is defined by
$$
(a_0+a_1X+\dots+a_nX^n)v=
a_0v+a_1f(v)+\dots+a_nf^n(v)
$$
For instance this approach can lead to a quite interesting insight on Jordan canonical form and to other invariants of the endomorphism $f$ that Cayley and Sylvester introduced with very different techniques.
