In algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, the elementary divisors 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...

 (PID) 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 M is isomorphic to a unique sum of the form

where and the are primary ideal

Primary ideal

In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n...

s.

The ideals are unique (up to order); the elements are unique up to associatedness, and are called the elementary divisors. Note that in a PID, primary ideals are powers of primes, so the elementary divisors . The nonnegative integer is called the free rank or Betti number of the module .

The elementary divisors 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.