Algebra representation
Polynomial algebras
### Polynomial algebras

Another important basic class of examples are representations of [polynomial algebras](/wiki/Polynomial_algebra "Polynomial algebra"), the free commutative algebras – these form a central object of study in [commutative algebra](/wiki/Commutative_algebra "Commutative algebra") and its geometric counterpart, [algebraic geometry](/wiki/Algebraic_geometry "Algebraic geometry"). A representation of a polynomial algebra in  variables over the [field](/wiki/Field_%28mathematics%29 "Field (mathematics)") *K* is concretely a *K*\-vector space with  commuting operators, and is often denoted K\[T\_1,\\dots,T\_k], meaning the representation of the abstract algebra K\[x\_1,\\dots,x\_k] where x\_i \\mapsto T\_i.

A basic result about such representations is that, over an [algebraically closed field](/wiki/Algebraically_closed_field "Algebraically closed field"), the representing [matrices](/wiki/Matrix_%28mathematics%29 "Matrix (mathematics)") are [simultaneously triangularisable](/wiki/Triangular_matrix%23Simultaneous_triangularisability "Triangular matrix#Simultaneous triangularisability").

Even the case of representations of the polynomial algebra in a single variable are of interest – this is denoted by K\[T] and is used in understanding the structure of a single [linear operator](/wiki/Linear_operator "Linear operator") on a [finite\-dimensional](/wiki/Dimension_%28vector_space%29 "Dimension (vector space)") vector space. Specifically, applying the [structure theorem for finitely generated modules over a principal ideal domain](/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain "Structure theorem for finitely generated modules over a principal ideal domain") to this algebra yields as [corollaries](/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain%23Corollaries "Structure theorem for finitely generated modules over a principal ideal domain#Corollaries") the various canonical forms of matrices, such as [Jordan canonical form](/wiki/Jordan_canonical_form "Jordan canonical form").

In some approaches to [noncommutative geometry](/wiki/Noncommutative_geometry "Noncommutative geometry"), the free noncommutative algebra (polynomials in non\-commuting variables) plays a similar role, but the analysis is much more difficult.


What is a corollary of applying the structure theorem for finitely generated modules over a principal ideal domain to this algebra?