Invariant factor

The invariant factors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain. If [math]\displaystyle{ R }[/math] is a PID and [math]\displaystyle{ M }[/math] a finitely generated [math]\displaystyle{ R }[/math]-module, then

[math]\displaystyle{ M\cong R^r\oplus R/(a_1)\oplus R/(a_2)\oplus\cdots\oplus R/(a_m) }[/math]

for some integer [math]\displaystyle{ r\geq0 }[/math] and a (possibly empty) list of nonzero elements [math]\displaystyle{ a_1,\ldots,a_m\in R }[/math] for which [math]\displaystyle{ a_1 \mid a_2 \mid \cdots \mid a_m }[/math]. The nonnegative integer [math]\displaystyle{ r }[/math] is called the free rank or Betti number of the module [math]\displaystyle{ M }[/math], while [math]\displaystyle{ a_1,\ldots,a_m }[/math] are the invariant factors of [math]\displaystyle{ M }[/math] and are unique up to associatedness.

The invariant factors of a matrix over a PID occur in the Smith normal form and provide a means of computing the structure of a module from a set of generators and relations.

See also

• Elementary divisors

References

• B. Hartley; T.O. Hawkes (1970). Rings, modules and linear algebra. Chapman and Hall. ISBN 0-412-09810-5.  Chap.8, p.128.
• Chapter III.7, p.153 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0