From 92a8f27cd2a1220c8a9114fde99d5a1475427e10 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Thu, 25 Dec 2025 13:02:49 +0100 Subject: [PATCH] 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 | 10 + .github/workflows/typos.yml | 21 + .gitignore | 1 + README.md | 1 + commutative-algebra-notes.pdf | Bin 0 -> 313487 bytes commutative-algebra-notes.tex | 1007 +++++++++++++++++++++++++++++++++ galois-theory-notes.pdf | Bin 346331 -> 346340 bytes galois-theory-notes.tex | 10 +- notes_fri_stir.pdf | Bin 240319 -> 240300 bytes notes_fri_stir.tex | 9 +- 10 files changed, 1050 insertions(+), 9 deletions(-) create mode 100644 .github/workflows/typos.toml create mode 100644 .github/workflows/typos.yml create mode 100644 commutative-algebra-notes.pdf create mode 100644 commutative-algebra-notes.tex diff --git a/.github/workflows/typos.toml b/.github/workflows/typos.toml new file mode 100644 index 0000000..19ff2b7 --- /dev/null +++ b/.github/workflows/typos.toml @@ -0,0 +1,10 @@ +[default.extend-words] +thm = "thm" + +# equations stuff +ba = "ba" +nd = "nd" + +# names +Strang = "Strang" +Bootle = "Bootle" diff --git a/.github/workflows/typos.yml b/.github/workflows/typos.yml new file mode 100644 index 0000000..de7bc65 --- /dev/null +++ b/.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 + diff --git a/.gitignore b/.gitignore index 552c4ca..d45d44d 100644 --- a/.gitignore +++ b/.gitignore @@ -12,3 +12,4 @@ *.snm *.vrb galois-theory-notes.bib +commutative-algebra-notes.bib diff --git a/README.md b/README.md index bc33bcd..e383c72 100644 --- a/README.md +++ b/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: diff --git a/commutative-algebra-notes.pdf b/commutative-algebra-notes.pdf new file mode 100644 index 0000000000000000000000000000000000000000..23403c0e9a8aecab0278208398bcc5b5d3779dd5 GIT binary patch literal 313487 zcmb5!LzFJSwk_ziZQK6Swr$(CZ5um{o!)8Nwr$&X)~PqDs&h}{R&*ko(VO3zW05P0 ziqiua*kH)#mWJ10*x87fh#ZWqVR(6A7-h}uEnF>$02}}&qW^PX7{#q@UCo?{7{zUk zT+KwyOdL$jVEFlATwI;ajO<`MH?}pk95%&~{AX+DRAhip%pDkXo`|RCJLH0i9D@qt z3Dr^TkoSobRqcL1XRa8irAy5^fpMA`7xEe#cYR&x&abS+EqZy25WjwXU#{M9lcqF< zIaQq9-MJ2zkaA~ORYZ$@-CMG9B^WOoa$NB3cs7p%jA*k*gCEYsoYLrVE_u|p&91)= zt~ija%wv;OspCmdrHhdoGjNTSPsBD#8#a&K0SR6p3x=!s$9cU<<5P*MHf0$UITVeW zRK`s&!uoU@HT9D>J=fk%eCDyJq@1i^O1$?dJ3P52HJ(fBj4t+tRS(;jZu+J@>+YM& zcKUknd61;|eN0N47=4`mwDaXvn>B6EEa`H3;cRU^lMYIOdPO_(goD5Y;=uJwm(Tcu 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+\begin{defn}{idempotent} + $e \in A$ such that $e^2=e$. +\end{defn} + +\begin{defn}{radical of an ideal} + $$rad I = \{ f \in A | f^n \in I~ \text{for some} n \}$$ + + $rad I$ is an ideal. + + $nilrad A = rad 0$ + + $rad I = \bigcap_{\substack{P \in \operatorname{Spec}(A)\\ P \supset I}} P$ +\end{defn} + +\begin{defn}{local ring} + A \emph{local ring} has a unique maximal ideal. + + Notation: local ring $A$, its maximal ideal $\mM$, residue field $K=A/\mM$: + $$A \supset \mM ~\text{or}~ (A, \mM) ~\text{or}~ (A, \mM, K)$$ +\end{defn} + +\subsection{$\mathbb{Z}$ and $K[X]$, two Principal Ideal Domains} + + +\begin{lemma}{} + $\mathbb{Z}$ is a PID. +\end{lemma} +\begin{proof} + Let $I$ a nonzero ideal of $\mathbb{Z}$. + + Since $I \neq \{0\}$, there is at least one nonzero integer in $I$. Choose the smallest element of $I$, namely $d$. + + Observe that $(d) \subseteq I$, since $d \in I$. Then, every multiple $nd \in I$, since $I$ is an ideal. + + Take $a \in I$. By the Euclidean division algorithm in $\mathbb{Z}$, $a=qd+r$, with $q,r \in \mathbb{Z}$ and $0 \leq r \leq d$. + + Then $r = a - qd \in I$, but $d$ was chosen to be the smallest positive element of $I$, so the only possibility is $r=0$. + + Hence, $a=qd$, so $a \in (d)$, giving $I \subseteq (d)$. + + Since we had $(d) \subseteq I$ and now we got $I \subseteq (d)$, we have $I = (d)$, so every ideal of $\mathbb{Z}$ is principal. Thus $\mathbb{Z}$ is a Principal Ideal Domain(PID). +\end{proof} + +\begin{lemma}{} + $K[X]$ is a PID. +\end{lemma} +\begin{proof} + This proof follows very similarly to the previous proof.\\ + + Let $K$ be a field, $K[X]$ a polynomial ring. + + Take $\{0\} \neq I \subseteq K[X]$. + + Since $I \neq \{0\}$, there is at least one non-zero polynomial in $I$. + + Let $p(X) \in I$ be of minimal degree among nonzero elements of $I$. + + Observe that $(p(X)) \subseteq I$, because $p(X) \in I$ and $I$ is an ideal. + + Let $f(X) \in I$. By Euclidean division algorithm in $K[X]$, $\exists q, r \in K[X]$ such that $f(X) = q(X) \cdot p(X) + r(X)$ with eithr $r(X)=0$ or $deg(r) < deg(p)$. + + Since $f,p \in I$, then $r(X) = f(X) - q(X)\cdot p(X) \in I$ + + If $r(X) \neq 0$, then $deg(r) < deg(p)$, which contradicts the minimality of $deg(p)$ in $I$. + + Therefore, $r(X)=0$, thus $f(X)=q(X)\cdot p(X)$, hence $f(X) \in (p(X))$. + Henceforth, $I \subseteq (p(X))$. + + Then, since $(p(X)) \subseteq I$ and $I \subseteq (p(X))$, we have that $I = (p(X))$. + + So every ideal of $K[X]$ is principal; thus $K[X]$ is a PID. + +\end{proof} + + + +\subsection{Lemmas, propositions and corollaries} + +Let $\Sigma$ be a partially orddered set. 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\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.\\ diff --git a/notes_fri_stir.pdf b/notes_fri_stir.pdf index 4f16693522464b477b4074cc3c5b05c29c2dd910..4eb5c83308f4fbee6ee84a52993c97e8e41e88d3 100644 GIT binary patch delta 11299 zcmai&Lv)}Glx<_%HY>Jm+qPXXzNBK?wyjE1v2Ckj+oNG)rgWFv=Txg!w?S`srP zYPi!CQ2Kak|NvXq=Z3%nj9E@z5J zs@jzPwU>omNO`^a)?S?sIsf!P+-1(|a>AYDI`TQluaf&Ny9Eq)=&iD=bbRJx#EW2u zBhX;gxaZR&*Aa`NTEMGXv2|Wy1r^HFj}-n5f3NGmw?Ew5YNVO)5{)llq?zcNd=I3Q 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