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