Module (mathematics) - Examples

Examples

• If K is a field, then the concepts "K-vector space" (a vector space over K) and K-module are identical.
• The concept of a Z-module agrees with the notion of an abelian group. That is, every abelian group is a module over the ring of integers Z in a unique way. For n > 0, let nx = x + x + ... + x (n summands), 0x = 0, and (−n)x = −(nx). Such a module need not have a basis—groups containing torsion elements do not. (For example, in the group of integers modulo 3, one cannot find even one element which satisfies the definition of a linearly independent set since when an integer such as 3 or 6 multiplies an element the result is 0. However if a finite field is considered as a module over the same finite field taken as a ring, it is a vector space and does have a basis.)
• 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. Hence when n = 1, R is an R-module, where the scalar multiplication is just ring multiplication. The case n = 0 yields the trivial R-module {0} consisting only of its identity element. Modules of this type are called free and if R has invariant basis number (e.g. any commutative ring or field) the number n is then the rank of the free module.
• If S is a nonempty set, M is a left R-module, and MS is the collection of all functions f : S → M, then with addition and scalar multiplication in MS defined by (f + g)(s) = f(s) + g(s) and (rf)(s) = rf(s), MS is a left R-module. The right R-module case is analogous. In particular, if R is commutative then the collection of R-module homomorphisms h : M → N (see below) is an R-module (and in fact a submodule of NM).
• If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring C∞(X). The set of all smooth vector fields defined on X form a module over C∞(X), and so do the tensor fields and the differential forms on X. More generally, the sections of any vector bundle form a projective module over C∞(X), and by Swan's theorem, every projective module is isomorphic to the module of sections of some bundle; the category of C∞(X)-modules and the category of vector bundles over X are equivalent.
• The square n-by-n matrices with real entries form a ring R, and the Euclidean space Rn is a left module over this ring if we define the module operation via matrix multiplication.
• 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.
• If R is a ring, we can define the ring Rop which has the same underlying set and the same addition operation, but the opposite multiplication: if ab = c in R, then ba = c in Rop. Any left R-module M can then be seen to be a right module over Rop, and any right module over R can be considered a left module over Rop.
• There are modules of a Lie algebra as well.