Home > Module (mathematics)
AlgebraIn abstract algebra, a module is a generalization of a vector space. In a vector space the set of scalars forms a field whereas in a module the scalars just form a ring. Much of the theory of modules consists of recovering desirable properties of vector spaces in the realm of modules over certain rings. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis.
1 Definition
Specifically, a left module over the ring R consists of an abelian group (M, +) and an operation R × M -> M (scalar multiplication, usually just written by juxtaposition, i.e. as rx for r in R and x in M) such that
For all r,s in R, x,y in M, we have
- (rs)x = r(sx)
- (r+s)x = rx+sx
- r(x+y) = rx+ry
- 1x = x
Usually, we simply write "a left R-module M" or RM.
Some authors omit condition 4 for the general definition of left modules, and call the above defined structures "unital left modules". In this encyclopedia however, all modules are assumed to be unital, and all rings are assumed to have a one.
A right R-module M or MR is defined similarly, only the ring acts on the right, i.e. we have a scalar multiplication of the form M × 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 left R-modules are the same as right R-modules and are simply called R-modules.
2 Examples
- If K is a field, then the concepts "K- vector space" and K-module are identical.
- Every abelian group M is a module over the ring of integers Z if we define nx = x + x + ... + x (n summands) for n > 0, 0x = 0, and (-n)x = -(nx) for n < 0.
- If R is any ring and n a natural number, then the cartesian product Rn is both a left and a right module over R if we use the component-wise operations. The case n=0 yields the trivial R-module {0} consisting only of its identity element.
- If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring R. The set of all smooth vector fields defined on X form a module over R, and so do the tensor fieldDifferential geometry Differential topology In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. It is used in differential geometry and the theory of manifolds, in algebraic geometry, in generas and the differential formGentler (and longer) introduction We initially work in an open set in R n''. A 0-form is defined to be a smooth function f''. When we integrate a function f over an m dimensional subspace S of R n we write it as : Consider dx . dx for a moment as formal os on X.
- The square n-by-n matricesAbstract algebra Algebra Linear algebra In mathematics, a matrix (plural matrices is a rectangular table of numbers or, more generally, of elements of a fixed ring. In this article, if unspecified, the entries of a matrix are always real or complex number with real entries form a ring R, and the Euclidean spaceEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers Rn is a left module over this ring if we define the module operation via matrix multiplicationThis article gives an overview of the various ways to multiply matrices. The Einstein notation is used throughout. Ordinary matrix product By far the most important way to multiply matrices is the usual matrix multiplication. It is defined between two mat.
- If R is any ring and I is any left ideal in R, then I is a left module over R. Analogously of course, right ideals are right modules.
