Ring theory is an algebraic subject which studies algebraic structures (sets with operations) having two operations extending and generalizing the concepts of fields such as real , complex and rational field. Module theory is an algeraic subject also and is cinsidered as a generalization of vector spaces over field where fields are replaced by rings.

One can think of a module as an abelian group acted upon by a ring of homomorphisms of that group into itself. Thus, for example, an ordinary vector space (an example of a module) is a group of vectors acted upon by the ring formed by the homomorphisms of the form v--->c*v for some scalar.

In one hand, Module Theory unifies (and generalizes) the study of Abelian Groups (which are the modules over the ring of integers), and the vector spaces (which are modules over fields). On the other hand, a (left or right ) ideal in a ring $R$ (and so $R$ itself) is just an instance of a module over $R$, thus studying the ring $R$ amounts to studying modules over it.