---
# System prepended metadata

title: 'Modules [ch.11]'
tags: [abstract algebra]

---

# 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**
![Screenshot 2024-11-16 at 9.55.44 AM](https://hackmd.io/_uploads/HyTQRPrzyg.png)
- **Definition 11.2: R - linear transformation, R -  module homomorphism**
![Screenshot 2024-11-16 at 9.59.12 AM](https://hackmd.io/_uploads/Bkb6CDHMkl.png)

## 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)**
![Screenshot 2024-11-18 at 12.42.47 AM](https://hackmd.io/_uploads/S1LZJ9DfJg.png)
- **Proposition 11.11: Quotient Group**
![Screenshot 2024-11-18 at 12.29.17 AM](https://hackmd.io/_uploads/BJYMkcPGyl.png)
- **Proposition 11.12:**
![Screenshot 2024-11-18 at 12.33.18 AM](https://hackmd.io/_uploads/HJy02Ywz1e.png)
- **Definition 11.13: Product**
![Screenshot 2024-11-18 at 12.47.10 AM](https://hackmd.io/_uploads/rkyMg5wGyl.png)
- **Definition 11.15: principal & cyclic submodule**
![Screenshot 2024-11-18 at 12.49.16 AM](https://hackmd.io/_uploads/rkB3xqDfJl.png)
- **Proposition 11.18**
![Screenshot 2024-11-18 at 3.05.27 AM](https://hackmd.io/_uploads/HJWFlhDfke.png)
- **Definition: 11.19: (direct) product, direct sum**
![Screenshot 2024-11-18 at 3.13.03 AM](https://hackmd.io/_uploads/HyULfnvM1e.png)
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**
![Screenshot 2024-11-19 at 12.00.35 AM](https://hackmd.io/_uploads/Hkmj8RuGyl.png)
![Screenshot 2024-11-19 at 12.02.54 AM](https://hackmd.io/_uploads/ByHEv0dGye.png)
- **Definition 11.23: Free, finitely generated, cyclic R-modules**
![Screenshot 2024-11-19 at 12.04.24 AM](https://hackmd.io/_uploads/rkjTvR_zJl.png)
![Screenshot 2024-11-19 at 12.24.40 AM](https://hackmd.io/_uploads/Hkl3B2AOzyx.png)
Note: check textbook to see other examples
- **Theorem 11.27**
![Screenshot 2024-11-19 at 12.54.19 AM](https://hackmd.io/_uploads/Sk1HXyFGyx.png)
Note:
    - The rank of a free R-module is well-defined.
    - check theorem 4.18
- **Definition 11.28: Rank**
![Screenshot 2024-11-19 at 12.55.25 AM](https://hackmd.io/_uploads/rkgtmJYfkg.png)
Note: Sometimes people use dimension as a synonym for rank.

## 11.5. Homomorphisms, Endomorphisms, Matrices
- **Definition 11.29: Space of Homomorphisms**
![Screenshot 2024-11-19 at 1.09.59 AM](https://hackmd.io/_uploads/S1bZwyFGkx.png)
- **Proposition 11.30**
![Screenshot 2024-11-19 at 1.09.45 AM](https://hackmd.io/_uploads/HJkQw1tMyg.png)
- **Definition: 11.31: Matrix $Mat_{r\times s}(R)$**
![Screenshot 2024-11-19 at 1.13.54 AM](https://hackmd.io/_uploads/SJWfdyYM1e.png)
- **Proposition 11.32**
![Screenshot 2024-11-19 at 1.15.50 AM](https://hackmd.io/_uploads/HyLHOkYzke.png)
- **Definition 11.33**
![Screenshot 2024-11-19 at 1.31.25 AM](https://hackmd.io/_uploads/Syie3JKf1e.png)
- **Proposition 11.34**
![Screenshot 2024-11-19 at 2.26.04 AM](https://hackmd.io/_uploads/BJyadgtGkl.png)

## 11.6. Noetherian Rings and Modules
- **Proposition 11.35: Submodule stablizes, Maxiaml Element**
![Screenshot 2024-11-20 at 12.32.08 AM](https://hackmd.io/_uploads/ryQckV9Mkl.png)
- **Definition 11.36, 11.37: Noetherian R-modules, Noetherian Rings**
![Screenshot 2024-11-20 at 12.32.14 AM](https://hackmd.io/_uploads/BymqyVczkx.png)
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**
![Screenshot 2024-11-19 at 1.37.37 PM](https://hackmd.io/_uploads/SyeE8qKfJl.png)
![Screenshot 2024-11-19 at 1.40.53 PM](https://hackmd.io/_uploads/HkIywqFfJg.png)
- **Proposition 11.41**
![Screenshot 2024-11-19 at 1.40.24 PM](https://hackmd.io/_uploads/r170IctGye.png)
- **Theorem 11.42**
![Screenshot 2024-11-19 at 11.43.21 PM](https://hackmd.io/_uploads/SyLmEQqGke.png)
- **Theorem 11.43: Hilbert Basis Theorem**
![Screenshot 2024-11-19 at 11.54.55 PM](https://hackmd.io/_uploads/SyZ08Qcfkx.png)
![Screenshot 2024-11-20 at 12.55.42 AM](https://hackmd.io/_uploads/SyxfB4qMkg.png)
## 11.7. Matrices with Entries in a Euclidean Domain
- **Definition 11.45: Smith Normal Form**
![Screenshot 2024-11-20 at 12.57.02 AM](https://hackmd.io/_uploads/H1xPBEqMJx.png)
- **Definition 11.46: Elementary Matrix Operations**
![Screenshot 2024-11-20 at 12.59.06 AM](https://hackmd.io/_uploads/SJs0SV9M1e.png)
- **Lemma 11.47: Diagonalization over Fields**
![Screenshot 2024-11-20 at 1.00.55 AM](https://hackmd.io/_uploads/H1OqLE5MJe.png)
- **Lemma 11.49: Diagonalization over Euclidean Domain**
![Screenshot 2024-11-20 at 1.03.50 AM](https://hackmd.io/_uploads/SJY-P4qM1g.png)

## 11.8. Finitely Generated Modules over Euclidean Domains
- **Theorem 11.50: Structure Theorem for Finitely Generated Modules over Euclidean Domains: Version 1**
![Screenshot 2024-11-20 at 2.01.03 AM](https://hackmd.io/_uploads/BkewNr9Myx.png)
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 $M$**
![Screenshot 2024-11-20 at 2.04.27 AM](https://hackmd.io/_uploads/ry-NBBqfyl.png)
- **Definition 11.53: Torsion Submodule, Anniliator Ideal**
![Screenshot 2024-11-20 at 11.51.41 PM](https://hackmd.io/_uploads/rkQ1_dif1l.png)
![Screenshot 2024-11-21 at 1.06.50 AM](https://hackmd.io/_uploads/BkuEYYjzyx.png)
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**
![Screenshot 2024-11-21 at 11.33.06 PM](https://hackmd.io/_uploads/Bke6463zJg.png)
- **Corollary 11.55**
![Screenshot 2024-11-22 at 12.15.24 AM](https://hackmd.io/_uploads/BkFsCa2fJe.png)

## 11.9. Applications of the Structure Theorem
- **Theorem 11.56: Structure Theorem for Finitely Generated Abelian Groups**
![Screenshot 2024-11-22 at 1.47.43 AM](https://hackmd.io/_uploads/HJMHVkTzkx.png)
- **Corollary 11.57**
![Screenshot 2024-11-22 at 1.48.21 AM](https://hackmd.io/_uploads/HyguVkTz1g.png)
- **Example 11.58**
![Screenshot 2024-11-22 at 1.52.19 AM](https://hackmd.io/_uploads/ByR8SkpMJg.png)
- **Definition 11.59: Jordan Matrix (Jordan Block)**
![Screenshot 2024-11-23 at 1.32.14 AM](https://hackmd.io/_uploads/H1VQGN0zyl.png)
Note: check eigenvectors in Chapter 10.
- **Theorem 11.60: Jordan Normal Form Theorem**
![Screenshot 2024-11-23 at 2.14.10 AM](https://hackmd.io/_uploads/Bkcl3V0z1l.png)
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.