From 25a8538245bda4e8224e7c8f3ee56c991ef5ace1 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Tue, 23 Dec 2025 19:25:03 +0100 Subject: [PATCH] add ideals related definitions --- commutative-algebra-notes.pdf | Bin 275717 -> 282885 bytes commutative-algebra-notes.tex | 94 ++++++++++++++++++++++++++++++++-- 2 files changed, 89 insertions(+), 5 deletions(-) diff --git a/commutative-algebra-notes.pdf b/commutative-algebra-notes.pdf index 9af0ba36347da67d41d83e1eb29b61bbbc719dba..36759e4322c33da03e6319b375dd831c0d21f830 100644 GIT binary patch delta 134144 zcmZs?V{qq9^z9qlwry*oiJggUXJY#s=NC+D+x8?ACllMY?TPR6Kj+@Mb*fIk+Hbn6 z_KV&1>9tmeouP#lqN7v)kdR_#XW>Smo>>@LLEvFaK*xmSVdGAk#iau_wdLd1dCwtU<-ukaH^d0c^w{6kimy1YB$YcJeK(IA2wii%4)i=458F0w8yk2^3ovh>H-Wcy zGc-|o%bl!RlGo*i8>}d#RwOR8sydCt?%o#KK^CsT9rMY1vru%fWXJ%p`>@tK;H}CvmW*e*S5Ags z_OehGPxjm*yJ&K$!M%hTa=7UOhipzfZJ(@?{LwBLz+uot>~wbP>It@aCtSJgzAwp> zo$7iXpt0=LSE>sUB9IS%6YGVAO~oV|s_yS0+fG$Fjg*a7;elV336HaHvuh+HFDDJN 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z*Dxl{fD5m{%GJw{z08Wh$5v(y!mFw#@kP;@7soQby_J1dO~~Ahb4`!Gjo-aA39ru% zpEe3}XFdu}ejeHpm*L&L4ak2LjplA@bM3Spvi59H3jp2K7wZmZ3>PC0oyMnbz3|a5 zYbk^T6#Vhz-$th_-th|=Am3FL2hR9z d{t2GvSQ8CeR{;0$. +\end{defn} +\begin{defn}{nilrad A} + set of all nilpotent elements of $A$; is an ideal of $A$. + + if $nilrad A = 0 ~\Longrightarrow$ $A$ has no nonzero nilpotents. + + + $$nilrad A = \bigcap_{P \in Spec(A)} P$$ +\end{defn} + +\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: locall 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{Lemmas, propositions and corollaries} \begin{thm}{AM.1.X}{Zorn's lemma} \label{zorn} @@ -339,6 +423,7 @@ $A[\psi] \subset End M$ is the subring geneerated by $A$ and the action of $\psi \begin{proof} By \ref{2.5}: since $\aA M = M$, we have $x M =0$ for some $x \equiv 1 \pmod {Jac(A)}$. (notice that at \ref{2.5} is $\pmod \aA$ but here we use $\pmod {Jac(A)}$, since we have $\aA \subseteq Jac(A)$). + (recall \ref{1.9}: $x \in Jac(A)$ iff $(1 - xy)$ is a unit in $A$, $\forall y \in A$).\\ By \ref{1.9}, $x$ is a unit in $A$ (thus $x^{-1}\cdot x=1$). Hence $M = x^{-1} \cdot \underbrace{x~ \cdot M}_{=0~ \text{(by \ref{2.5})}} = 0$. @@ -381,7 +466,7 @@ $A[\psi] \subset End M$ is the subring geneerated by $A$ and the action of $\psi $$\frac{\aA M + N}{N} = \left\{ y + N ~~|~~ y \in \aA M +N \right\} = \aA M + N$$ thus every $y \in \aA M +N$ can be written as - $$y=x+n,~~ \text{with} x \in \aA M,~ n\in M$$ + $$y=x+n,~~ \text{with} x \in \aA M,~ n\in N$$ which comes from \eqref{eq:2.7.1}. Thus, $y + N = (x+n)+N = x+N$, since $n \in N$ is zero in the quotient. @@ -500,9 +585,8 @@ As in with rings, it is equivalent to say that \end{proof} \begin{lemma}{R.3.4.L} - for submodules $M_1 \subset M_2 \subset M$,\\ - - $L \cap M_1 = L \cap M_2$ and $\beta(M_1) = \beta(M_2) ~\Longrightarrow~ M_1 = M_2$. + for submodules $M_1 \subset M_2 \subset M$, + $$L \cap M_1 = L \cap M_2 ~\text{and}~ \beta(M_1) = \beta(M_2) ~\Longrightarrow~ M_1 = M_2$$ \end{lemma} \begin{proof} if $m\in M_2$, then $\beta(m) \in \beta(M_1) = \beta(M_2)$, so that there is an $n \in M_1$ such that $\beta(m)=\beta(m)$.