Generally, something that is modular is constructed so as to facilitate easy assembly, flexible arrangement, and/or repair.
In abstract algebra, a left Rmodule consists of an abelian group (M, +) together with a ring of scalars (R,+,*) and an operation R x M > M (scalar multiplication, usually just written by juxtaposition, i.e. as rx for r in R and x in M) such that
Usually, we simply write "a left Rmodule M" or _{R}M.
If furthermore, R has an identity 1 and for all x in M, 1x = x, then it is called a unital module.
A right Rmodule M or M_{R} is defined similarly, only the ring acts on the right, i.e. we have a scalar multiplication of the form M x R > M, and the above three axioms are written with scalars r and s on the right of x and y. If R is commutative, then the left Rmodule is the same as the right Rmodule and is simply called an Rmodule.
If R is a field, then an Rmodule is also called a vector space. Modules are thus generalizations of vector spaces, and much of the theory of modules consists of recovering desirable properties of vector spaces in the realm of modules over certain rings. However, in general, an Rmodule may not have a basis[?].
Suppose M is an Rmodule and N is a subgroup of M. Then N is a submodule (or Rsubmodule, to be more explicit) if, for any n in N and any r in R, the product rn is in N (or nr for a right module).
If M and N are Rmodules, then a map f : M > N is a homomorphism if, for any m, n in M and r, s in R, f(rm + sn) = rf(m) + sf(n). This, like any homomorphism of mathematical objects, is just a mapping which preserves the structure of the objects.
If M is a left Rmodule, 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 M. The set of all group endomorphisms of M is denoted End_{Z}(M) and forms a ring under addition and composition, and sending a ring element to its action actually defines a ring homomorphism from R to End_{Z}(M).
Such a ring homorphism R → End_{Z}(M) is called a representation of R over the abelian group M; an alternative and equivalent way to defining left Rmodules is to say that a left Rmodule is an abelian group M together with a representation of R over it.
A representation is called faithful if and only if the map R → End_{Z}(M) is onetoone. Every abelian group is a module over the integers, and is either faithful under them or some modular arithmetic.
Search Encyclopedia

Featured Article
