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 $R$ is a PID and $M$ a finitely generated $R$ -module, then

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.

单词	Invariant factor
释义	Invariant factor 英语百科 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 $R$ is a PID and $M$ a finitely generated $R$ -module, then 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.
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