The invariant factors of a module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

 over a principal ideal domain

Principal ideal domain

In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...

 occur in one form of the structure theorem for finitely generated modules over a principal ideal domain

Structure theorem for finitely generated modules over a principal ideal domain

In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules can be uniquely decomposed in...

.

If is a PID

Principal ideal domain

In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...

 and a finitely generated -module, then


for some and nonzero elements for which . The nonnegative integer is called the free rank or Betti number of the module , while are the invariant factors of and are unique up to associatedness.

The invariant factors of a matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 over a PID occur in the Smith normal form

Smith normal form

In mathematics, the Smith normal form is a normal form that can be defined for any matrix with entries in a principal ideal domain . The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible square matrices...

and provide a means of computing the structure of a module from a set of generators and relations.