[1] Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication.
In commutative algebra, both ideals and quotient rings are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules.
Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "well-behaved" ring, such as a principal ideal domain.
However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis, and, even for those that do (free modules), the number of elements in a basis need not be the same for all bases (that is to say that they may not have a unique rank) if the underlying ring does not satisfy the invariant basis number condition, unlike vector spaces, which always have a (possibly infinite) basis whose cardinality is then unique.
Often the symbol · is omitted, but in this article we use it and reserve juxtaposition for multiplication in R. One may write RM to emphasize that M is a left R-module.
A right R-module MR is defined similarly in terms of an operation · : M × R → M. Authors who do not require rings to be unital omit condition 4 in the definition above; they would call the structures defined above "unital left R-modules".
Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements.
The kernel of a module homomorphism f : M → N is the submodule of M consisting of all elements that are sent to zero by f, and the image of f is the submodule of N consisting of values f(m) for all elements m of M.[4] The isomorphism theorems familiar from groups and vector spaces are also valid for R-modules.
If M is a left R-module, then the action of an element r in R is defined to be the map M → M that sends each x to rx (or xr in the case of a right module), and is necessarily a group endomorphism of the abelian group (M, +).
Such a ring homomorphism R → EndZ(M) is called a representation of R over the abelian group M; an alternative and equivalent way of defining left R-modules is to say that a left R-module is an abelian group M together with a representation of R over it.
With this understanding, a left R-module is just a covariant additive functor from R to the category Ab of abelian groups, and right R-modules are contravariant additive functors.
These form a category OX-Mod, and play an important role in modern algebraic geometry.
If X has only a single point, then this is a module category in the old sense over the commutative ring OX(X).
This allows a further generalization of the concept of vector space incorporating the semirings from theoretical computer science.