ABSTRACT
The structure theorem for finitely-generated abelian groups and Jordan canonical form for endomorphisms of finite-dimensional vector spaces are example corollaries of a common idea.
Let R be a principal ideal domain, that is, a commutative ring with identity such that every ideal I in R is principal, that is, the ideal can be expressed as
I = R · x = {r · x : r ∈ R} for some x ∈ R. An R-module M is finitely-generated if there are finitely-many m1, . . . ,mn in M such that every element m in M is expressible in at least one way as
m = r1 ·m1 + . . .+ rn ·mn with ri ∈ R. A basic construction of new R-modules from old is as direct sums: given R-modules M1, . . . ,Mn, the direct sum R-module
M1 ⊕ . . .⊕Mn
is the collection of n-tuples (m1, . . . ,mn) with mi ∈Mi, with component-wise operation [1]
(m1, . . . ,mn) + (m′1, . . . ,m ′ n) = (m1 +m
′ 1, . . . ,mn +m
and the multiplication [2] by elements r ∈ R by
r · (m1, . . . ,mn) = (rm1, . . . , rmn)
10.1.1 Theorem: Let M be a finitely-generated module over a PID R. Then there are uniquely determined ideals
I1 ⊃ I2 ⊃ . . . ⊃ It such that
M ≈ R/I1 ⊕R/I2 ⊕ . . .⊕R/It The ideals Ii are the elementary divisors of M , and this expression is the elementary divisor form of M . [3]
Proof: (next chapter) ///
The following proposition (which holds in more general circumstances) suggests variations on the form of the structure theorem above.
