Advanced Algebra II

Advanced Algebra II

modules over principal ideal domain

In order to prove the uniqueness of the decomposition and also an- other description by invariant factors. We need to work more.

Lemma 0.1. Let M, N be module over a principal ideal domain, r ∈ R and p ∈ R is prime.

(1) rM and M[r] := {x ∈ M|rx = 0} are submodules.

(2) M[p] is a vector space over R/(p).

(3) (R/(pⁿ))[p] ∼= R/(p) and p^m(R/(pⁿ)) ∼= R/(p^n−m) for m < n.

(4) If M ∼= ⊕Mi, then rM ∼= ⊕rMi and M[r] ∼= ⊕Mi[r].

(5) If f : M → N a _RM-isomorphism, then f : M_τ ∼= Nτ, and f : M(p) ∼= N(p).

Proof. The proof are straightforward. We leave it to the readers. ¤ Lemma 0.2. Let r =Q

i=1,...,kpⁿ_iⁱ. Then

R/(r) ∼= ⊕i=1,...,kR/(pⁿ_iⁱ).

Proof. The is a generalization of Chinese Remainder Theorem. Just

copy the proof then we are done. ¤

Theorem 0.3. Let M be a finitely generated module over a principal ideal domain R. Then

(1) M is a direct sum of a free module F of finite rank and a finite number of cyclic torsion modules of order r₁, ..., r_m respectively.

Where r₁|r₂|...|r_m. And the rank of F and the list of ideals (r₁), ..., (r_m) is unique.

(2) M is a direct sum of a free module F of finite rank and a fi- nite number of cyclic torsion modules of order p^a₁¹, ..., p^a_k^k respec- tively. And the rank of F and the list of ideals (p^a₁¹), ..., (p^a_k^k) is unique.

The r₁, ..., r_m are called invariant factors. And p^a₁¹, ..., p^a_k^k are called elementary divisors.

Proof. We have seen the existence of decomposition into elementary divisors. By the similar method as we did in finitely generated abelian groups, one can construct the invariant factors out of elementary divi- sors. Thus it suffices to prove uniqueness for both cases.

For a fix prime p, we consider M[p]. It’s a vector space over R/(p) and d₁ := dim_R/(p)M[p] measure the number of elementary divisor of order pⁿ. We next consider pM[p] and define d₂ := dim_R/(p)pM [p].

Then d₂ measure the number of elementary divisor of order pⁿ with n ≥ 2. Inductively, we define d_k := dim_R/(p)p^k−1M[p] which measure the number of elementary divisor of order pⁿ with n ≥ k. The point is that these d_k are uniquely determined by M. It’s easy to see that M

1

2

has d_k− d_k+1 elementary divisor (p^k), and this is uniquely determined

by M. ¤

Corollary 0.4 (Jordan canonical form). Let A be a n × n matrix over a field k. Suppose that it satisfies a polynomial (in particular, char- acteristic polynomial or minimal polynomial) which splits into linear factors. Then A has Jordan canonical form.

Proof. We view A as a linear transformation on a n-dimensional vector space V over k. Then V can be viewed as a k[x]-module by f (x)v :=

f (A)v. It’s well-known that k[x] is a principal ideal domain. It’s clear that V is a torsion module since A satisfies a polynomial. Thus V = PV (x − λ), or even V decomposes into cyclic submodule of order (x − λ_i)^ki. It’s easy to check that submodule of V is a vector subspace.

And one can pick a basis in a cyclic submodule so that the matrix is

represented as a Jordan block. ¤

Corollary 0.5 (fundamental theorem of finitely generated abelian group).