notes-archive/docs/math/abstract-algebra/DF-12-modules-pids.md

Dummit & Foote Chapter 12 - Modules over Principal Ideal Domains

Section 12.1 The Basic Theory

Definition. The left $R$-module M is said to be a Noetherian $R$-module if there are no infinitely increasing chains of submodules. That is, given

M_1 \subseteq M_2 \subseteq \ldots

there exists some k \in \mathbb{N} such that given any n \in \mathbb{N} with n \geq k, then M_n = M_k.

Definition. A ring R is Noetherian if it is Noetherian when viewed as a left $R$-module over itself.

Theorem. Let R be a ring and M a left $R$-module. Then, the following are equivalent:

1. M is Noetherian
2. Every nonempty set of submodules of M contains a maximal element under inclusion
3. Every submodule of M is finitely-generated

Corollary. If R is a principal ideal domain (PID), then all nonempty set of ideals of R has a maximal element. Additionally, R is as Noetherian ring.

Proposition. Let R be an integral domain, and M be a free $R$-module of rank n < \infty. Then, given S is subset M with |S| > n, the elements of S are $R$-linearly dependent.

Definition. Given R an integral domain and M an $R$-module,

\text{Tor}(M) = \{ x \in M | rx = 0 \text{ for any } r \neq 0 \}

This is the torsion submodule of M. If \text{Tor}(M) is empty, then M is torsion-free.

Definition. Let R be an integral domain and M be an $R$-module. Then, given a submodule N,

\text{Ann}_R(N) = \{r \in R | rn = 0 \text{ for all } n \in N \}

This ideal of R is the annihilator of $N$. That is, \text{Ann}(N) is the set of elements of R such that (r)N = \{ 0 \}.

Note that if N is not a torsion submodule of M, then \text{Ann}(N) = (0)R. Additionally, given N, L are submodules of M with N \subseteq L, then \text{Ann}(N) \subseteq \text{Ann}(L).

Additionally, if R is a PID, as \text{Ann}_R(N) is an ideal, \text{Ann}(N) = (n)R and \text{Ann}(L) = (l)R for some n, l \in R such that n | l.

Definition. Given any integral domain R, the rank of an $R$-module M is the maximum number of $R$-linearly independent elements of M.

Corollary. The rank of a free module is the number of generating elements.

Theorem. Let R be a principal ideal domain, and M be a free $R$-module of finite rank m, and N be a submodule of M. Then,

1. N is a free submodule with rank n \leq m.
2. There exists a basis y_1, y_2, \ldots, y_m of M so that r_1 y_1, r_2 y_2, \ldots, r_m y_n is a basis of N for some r_i \in R and r_1 | r_2 | \ldots | r_n

Theorem. Fundamental Theorem, Existence: Invariant Form. Let R be a PID and M be a finitely generated $R$-module. THen,

• M is isomorphic for some r \in \mathbb{N}\cup{0}, a_1, \ldots, a_m \neq 0 \in R such that a_1 | a_2 | \ldots | a_m, with

M \cong R^{\oplus r} \oplus \frac{R}{(a_1)R} \oplus \frac{R}{(a_2)R} \oplus \ldots \oplus \frac{R}{(a_m)R}

• M is torsion-free if and only if M is free

• Note that

\text{Tor}{M} \cong \frac{R}{(a_1)R} \oplus \frac{R}{(a_2)R} \oplus \ldots \oplus \frac{R}{(a_m)R}

As a consequence, M is a torsion module if and only if r = 0.

Definition. In the above, r is the free rank of M, and a_1, \ldots, a_m are the invariant factors of M.

Theorem. Fundamental Theorem, Existence: Elementary Divisor Form. The sum above can be written as

M \cong R^{\oplus r} \oplus \frac{R}{(p_1^{\alpha_1})R} \oplus \frac{R}{(p_2^{\alpha_2})R} \oplus \ldots \oplus \frac{R}{(p_t^{\alpha_t})R}

with p_t non-unique primes and \alpha_t non-unique, but with (p_t^{\alpha_t}) unique. These are called the elementary divisors of M.

TODO: Incomplete for Now