# Modules [ch.11]
- Important Note: In this chapter, ring means commutative ring, and as usual, our rings always have a multiplicative identity element.

## 11.1. What Is a Module?
- **Definition 11.1: R-Modules**
- **Definition 11.2: R - linear transformation, R - module homomorphism**

## 11.2. Examples of Modules
- check the textbook for the examples.

## 11.3. Submodules and Quotient Modules
- **Definition 11.9: Submodule (span of a subgroup)**
- **Proposition 11.11: Quotient Group**
- **Proposition 11.12:**
- **Definition 11.13: Product**
- **Definition 11.15: principal & cyclic submodule**
- **Proposition 11.18**
- **Definition: 11.19: (direct) product, direct sum**
Note: [direct sum and direct product (chinese)](https://www.youtube.com/watch?v=JLOo7xMLjTE)

## 11.4. Free Modules and Finitely Generated Modules
- **Definition: 11.20**
- **Definition 11.23: Free, finitely generated, cyclic R-modules**
Note: check textbook to see other examples
- **Theorem 11.27**
Note:
- The rank of a free R-module is well-defined.
- check theorem 4.18
- **Definition 11.28: Rank**
Note: Sometimes people use dimension as a synonym for rank.

## 11.5. Homomorphisms, Endomorphisms, Matrices
- **Definition 11.29: Space of Homomorphisms**
- **Proposition 11.30**
- **Definition: 11.31: Matrix $Mat_{r\times s}(R)$**
- **Proposition 11.32**
- **Definition 11.33**
- **Proposition 11.34**

## 11.6. Noetherian Rings and Modules
- **Proposition 11.35: Submodule stablizes, Maxiaml Element**
- **Definition 11.36, 11.37: Noetherian R-modules, Noetherian Rings**
Note:
- [Definition of maximal submodule](https://en.wikipedia.org/wiki/Maximal_ideal#:~:text=For%20an%20R%2Dmodule%20A,%3D%20M%20or%20N%20%3D%20A.)
- The maximal element $N$ is finitely generated.
- This does not mean that every submodule in S is contained in the maximal element N. It only means that N is not properly contained in any other submodule in the collection S. See **Definition 14.14** for a generalization to arbitrary partially ordered sets.
- **Example 11.38, 11.39, 11.40: Noetherian ring examples**
- **Proposition 11.41**
- **Theorem 11.42**
- **Theorem 11.43: Hilbert Basis Theorem**
## 11.7. Matrices with Entries in a Euclidean Domain
- **Definition 11.45: Smith Normal Form**
- **Definition 11.46: Elementary Matrix Operations**
- **Lemma 11.47: Diagonalization over Fields**
- **Lemma 11.49: Diagonalization over Euclidean Domain**

## 11.8. Finitely Generated Modules over Euclidean Domains
- **Theorem 11.50: Structure Theorem for Finitely Generated Modules over Euclidean Domains: Version 1**
Note$^{15}$: Theorem 11.50 is true more generally if R is a PID. However, since most applications, including the ones in this book, are for Euclidean domains.
- **Definition 11.51: Rank of $M$, Elementary Divisor of $M$**
- **Definition 11.53: Torsion Submodule, Anniliator Ideal**
Note:
- For 11.19: for some $a \in R$, $M_{tors} \cdot a = 0$ only when $R = 0$ because $R$ is an Euclidean domain which is also a PID, so there is no zero divisor. This means $R^r \cdot a = 0$ only when $R\times R\times...\times R=0\times0\times ...\times 0$
- For $Ann(M_{tors})=b_sR$, $\,\,(R/b_1R\times R/b_2R\times ...\times R/b_sR) \cdot b_sR$ will be quotient to $0$.
- **Definition 11.54: Structure Theorem for Finitely Generated Modules over Euclidean Do- mains: Version 2**
- **Corollary 11.55**

## 11.9. Applications of the Structure Theorem
- **Theorem 11.56: Structure Theorem for Finitely Generated Abelian Groups**
- **Corollary 11.57**
- **Example 11.58**
- **Definition 11.59: Jordan Matrix (Jordan Block)**
Note: check eigenvectors in Chapter 10.
- **Theorem 11.60: Jordan Normal Form Theorem**
Note: $L$ is a linear operator, so $L$ may be $x, x+1, x+\lambda,...$. That why the author use proposition8.6(a) to prove this theorem.