The invariant factors of a

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...

over a

principal ideal domain In mathematics, 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 princip ...

(PID) occur in one form of the

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 finit ...

. If

R

is a PID and

M

a finitely generated

R

-module, then :

M\cong R^r\oplus R/(a_1)\oplus R/(a_2)\oplus\cdots\oplus R/(a_m)

for some integer

r\geq0

and a (possibly empty) list of nonzero elements

a_1,\ldots,a_m\in R

for which

a_1 \mid a_2 \mid \cdots \mid a_m

. The nonnegative integer

r

is called the ''free rank'' or ''Betti number'' of the module

M

, while

a_1,\ldots,a_m

are the ''invariant factors'' of

M

and are unique up to

associatedness In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...

. The invariant factors of a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

over a PID occur in the

Smith normal form In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can b ...

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

References

* Chap.8, p.128. * Chapter III.7, p.153 of Module theory {{Abstract-algebra-stub

See also

References