port initial notes on commutative algebra (ideals, modules, Noetherian rings) (#1)

* port initial notes on commutative algebra: ideals & modules, Nakayama's lemma, etc

* port notes on Noetherian rings&modules

* add ideals related definitions

* improve Cayley-Hamilton proof (specially determinant trick explanation)

* polishing

* add typos detection

* add some exercises, and proof of Z and K[X} being PID

.github/workflows/typos.toml

@@ -0,0 +1,10 @@
+[default.extend-words]
+thm = "thm"
+
+# equations stuff
+ba = "ba"
+nd = "nd"
+
+# names
+Strang = "Strang"
+Bootle = "Bootle"

.github/workflows/typos.yml

@@ -0,0 +1,21 @@
+name: typos
+on:
+  pull_request:
+    branches: [ main ]
+    types: [ready_for_review, opened, synchronize, reopened]
+  push:
+    branches:
+      - main
+
+jobs:
+  typos:
+    if: github.event.pull_request.draft == false
+    name: Spell Check with Typos
+    runs-on: ubuntu-latest
+    steps:
+    - uses: actions/checkout@v4
+    - name: Use typos with config file
+      uses: crate-ci/typos@master
+      with: 
+        config: .github/workflows/typos.toml
+

.gitignore

@@ -12,3 +12,4 @@
 *.snm
 *.vrb
 galois-theory-notes.bib
+commutative-algebra-notes.bib

README.md

@@ -7,6 +7,7 @@ Notes, code and documents done while reading books and papers.
 - [Notes on "Abstract Algebra" book, by Charles C. Pinter](abstract-algebra-charles-pinter-notes.pdf)
 - [Notes on Weil pairing](weil-pairing.pdf)
 - [Notes on Galois Theory](galois-theory-notes.pdf)
+- [Notes on Commutative Algebra](commutative-algebra-notes.pdf)
 
 
 In-between math & crypto:

galois-theory-notes.tex

@@ -277,7 +277,7 @@
 \end{thm}
 
 \begin{thm}{9.10}
-  An irreducible polynomial $f \in K[t]$ ($K \subseteq \mathbb{C}$) is \emph{separable over} $K$ if it has simple zeros in $\mathbb{C}$, or equivelently, simple zeros in its splitting field.
+  An irreducible polynomial $f \in K[t]$ ($K \subseteq \mathbb{C}$) is \emph{separable over} $K$ if it has simple zeros in $\mathbb{C}$, or equivalently, simple zeros in its splitting field.
 \end{thm}
 
 \begin{lemma}{9.13}
@@ -809,7 +809,7 @@
 
   Let $f_i$ be the minimal polynimal of $\alpha_i$ over $K$.
 
-  Then, $M \supseteq L$ is spliting field of $\Prod_{i=1}^r f_i$, since $M$ is normal enclosure of $L:K$.
+  Then, $M \supseteq L$ is splitting field of $\Prod_{i=1}^r f_i$, since $M$ is normal enclosure of $L:K$.
 
   For every zero $\beta_{ij}$ of $f_i$ in $M$,\\
   $\exists$ an isomorphism $\sigma: K(\alpha_i) \longrightarrow K(\beta_{ij})$ by Corollary \ref{5.13}.
@@ -1065,7 +1065,7 @@ For $n \geq 3, ~~\mathbb{D}_n \subseteq \mathbb{S}_n$ (subgroup of the Symmetry
   then, $\exists$ at least one $c$ in $(a,b)$ such that $Df(c)=0$.
 \end{thm}
 \begin{proof} (proof source: cue math website)
-  Notice that when $Df(x_i)=0$ occours, is a maximum or minimum (extrema) value of $f$.
+  Notice that when $Df(x_i)=0$ occurs, is a maximum or minimum (extrema) value of $f$.
   $\Longrightarrow$ if a function is continuous, it is guaranteed to have both a maximum and a minimum point in the interval.\\
 
   Two possibilities:
@@ -1076,9 +1076,9 @@ For $n \geq 3, ~~\mathbb{D}_n \subseteq \mathbb{S}_n$ (subgroup of the Symmetry
     since $f$ not constant, must change directions in ordder to start and end at the same $y$-value ($f(a)=f(b)$).\\
     Thus at some point between $a$ and $b$ it will either have a minimum, maximum or both.
     \begin{enumerate}[a.]
-      \item does the maximum occour at a point where $Df > 0$?\\
+      \item does the maximum occur at a point where $Df > 0$?\\
 	No, because if $Df > 0$, then $f$ is increasing, but it can not increase since we're at its maximum point.
-      \item does the maximum occour at a point where $Df < 0$?\\
+      \item does the maximum occur at a point where $Df < 0$?\\
 	No, because if $Df < 0$, then $f$ is deccreasing, which means that at our left it was larger, but we're at a maximum point, so a contradiction.
     \end{enumerate}
     Same with minimus.\\

notes_fri_stir.tex

@@ -76,7 +76,7 @@ Consider the following protocol:
 \end{enumerate}
 
 %/// TODO tabulate this next lines
-With high probablility, $\alpha$ will not cancel the coeffs with $deg \geq d+1$. % TODO check which is the name of this theorem or why this is true
+With high probability, $\alpha$ will not cancel the coeffs with $deg \geq d+1$. % TODO check which is the name of this theorem or why this is true
 
 Let $g(x)=a \cdot x^{d+1}, ~~ h(x)=b \cdot x^{d+1}$, and set $f(x) = g(x) + \alpha h(x)$.
 Imagine that P can chose $\alpha$ such that $a x^{d+1} + \alpha \cdot b x^{d+1} = 0$, then, in $f(x)$ the coefficients of degree $d+1$ would cancel.
@@ -314,14 +314,15 @@ $$g(z) = (f(z)-f(r))\cdot (z-r)^{-1}$$
 
 
 \section{STIR (main idea)}
-\emph{Update from 2024-03-22, notes from Héctor Masip Ardevol (\href{https://hecmas.github.io/}{https://hecmas.github.io}) explanations.}
+\emph{Update from 2024-03-22, notes from Héctor Masip Ardevol\\
+(\href{https://hecmas.github.io/}{https://hecmas.github.io}) explanations.}
 
 \vspace{0.3cm}
 Let $p \in \mathbb{F}[x]^{<n}$.
 
 In FRI we decompose $p(x)$ as
 $$p(x) = p_e(x^2) + x \cdot p_o(x^2)$$
-with $p_e, p_o \in \mahtbb{F}[x]^{<n}$ containing the even and odd powers respectively.
+with $p_e, p_o \in \mathbb{F}[x]^{<n}$ containing the even and odd powers respectively.
 
 The next FRI polynomial is
 $$p_1(x) = p_e(x) + \alpha p_o(x)$$
@@ -329,7 +330,7 @@ for $\alpha \in^R \mathbb{F}$.
 
 In STIR, this would be $q(x)=x^2$,
 $$Q(z,y) = p_e(y) + z \cdot p_o(y)$$
-and then, $p(x) = Q(x, q(x))$. And $Q$ fullfills the degree from Fact 4.6 from the STIR paper.
+and then, $p(x) = Q(x, q(x))$. And $Q$ fulfills the degree from Fact 4.6 from the STIR paper.
 
 We can generalize to a $q$ with bigger degree, or with another shape, and adapting $Q$ on the choice of $q$.