Invariant factor explained

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

R is a PID and M a finitely generated R -module, thenM\cong R ^{r ⊕} R/(a _{1) ⊕} R/(a _{2) ⊕ … ⊕} R/(a _{m)}

for some integer

r\geq 0 and a (possibly empty) list of nonzero elements a _{1,\ldots,a} _{m\in} R for which a _{1} \mid a _{2} \mid … \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.

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

• Book: Brian Hartley

. B. Hartley . Brian Hartley . T.O. Hawkes . Rings, modules and linear algebra . Chapman and Hall . 1970 . 0-412-09810-5 . Chap.8, p.128.

• Chapter III.7, p.153 of