If S is a subring of R then any R-module can be considered as an S-module by restricting scalar multiplication to S M. For example, a complex vector space can be considered as a real vector space …

Modules are a generalization of the vector spaces of linear algebra in which the \scalars" are allowed to be from an arbitrary ring, rather than a ̄eld. This rather modest weakening of the axioms is quite far …

“Module” will always mean left module unless stated otherwise. Most of the time, there is no reason to switch the scalars from one side to the other (especially if the underlying ring is commutative).

We’ll later see how to understand module by looking at generators and relations — this turns out to be easier than the corresponding problem for a group. But first we’ll look at another example of a …

sub-modules. Proposition 2. Given a sub-R-module S ⊂ M, the quotient abelian group: M/S = {m + S | m ∈ M}/ ∼ is an R-module wit product a(m + S) = am + S.

Before we deal with deeper results on the structure of rings with the help of module theory we want to provide elementary definitions and con-structions in this chapter.

Here we cover all the basic material on modules and vector spaces required for embarkation on advanced courses. Concerning the prerequisite algebraic background for this, we mention that any …