From bc2868bdb026ec73d97f397331da552953e22db9 Mon Sep 17 00:00:00 2001
From: Juan Marin Noguera <juan@mnpi.eu>
Date: Sat, 24 Sep 2022 20:12:53 +0200
Subject: Inicio tema 1 AC

---
 ac/n.lyx  |  177 ++++++
 ac/n1.lyx | 1989 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 2 files changed, 2166 insertions(+)
 create mode 100644 ac/n.lyx
 create mode 100644 ac/n1.lyx

(limited to 'ac')

diff --git a/ac/n.lyx b/ac/n.lyx
new file mode 100644
index 0000000..cfda71e
--- /dev/null
+++ b/ac/n.lyx
@@ -0,0 +1,177 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input{../defs}
+\usepackage[x11names, svgnames, rgb]{xcolor}
+%\usepackage[utf8]{inputenc}
+\usepackage{tikz}
+\usetikzlibrary{snakes,arrows,shapes}
+\end_preamble
+\use_default_options true
+\begin_modules
+algorithm2e
+\end_modules
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
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+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize 10
+\spacing single
+\use_hyperref false
+\papersize a5paper
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 0.2cm
+\topmargin 0.7cm
+\rightmargin 0.2cm
+\bottommargin 0.7cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
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+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle empty
+\listings_params "basicstyle={\ttfamily}"
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Álgebra conmutativa
+\end_layout
+
+\begin_layout Date
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+def
+\backslash
+cryear{2022}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "../license.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Bibliografía:
+\end_layout
+
+\begin_layout Itemize
+Alberto del Valle Robles.
+ 
+\emph on
+Álgebra Conmutativa Curso 2021–2022, Apuntes de Clase
+\emph default
+.
+ Cuarto curso del Grado en Matemáticas.
+ Departamento de Matemáticas, Universidad de Murcia.
+ Basado en apuntes previos de José Luis García Hernández.
+\backslash
+
+\end_layout
+
+\begin_layout Itemize
+Clases de Manuel Saorín Castaño.
+\end_layout
+
+\begin_layout Chapter
+Anillos conmutativos
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n1.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/ac/n1.lyx b/ac/n1.lyx
new file mode 100644
index 0000000..1f8cc1c
--- /dev/null
+++ b/ac/n1.lyx
@@ -0,0 +1,1989 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input{../defs}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Un 
+\series bold
+grupo abeliano
+\series default
+ es un par 
+\begin_inset Formula $(A,+)$
+\end_inset
+
+ formada por un conjunto 
+\begin_inset Formula $A$
+\end_inset
+
+ y una 
+\series bold
+suma
+\series default
+ 
+\begin_inset Formula $+:A\times A\to A$
+\end_inset
+
+ asociativa, conmutativa, con un elemento neutro 
+\begin_inset Formula $0\in A$
+\end_inset
+
+ llamado 
+\series bold
+cero
+\series default
+ y en el que cada 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ posee un simétrico u 
+\series bold
+opuesto
+\series default
+ 
+\begin_inset Formula $-a$
+\end_inset
+
+.
+ Un 
+\series bold
+anillo
+\series default
+ es una terna 
+\begin_inset Formula $(A,+,\cdot)$
+\end_inset
+
+ formada por un grupo abeliano 
+\begin_inset Formula $(A,+)$
+\end_inset
+
+ y un 
+\series bold
+producto
+\series default
+ 
+\begin_inset Formula $\cdot:A\times A\to A$
+\end_inset
+
+ asociativo y distributivo respecto a la suma (
+\begin_inset Formula $(a+b)\cdot c=(a\cdot c)+(b\cdot c)$
+\end_inset
+
+ y 
+\begin_inset Formula $c\cdot(a+b)=(c\cdot a)+(c\cdot b)$
+\end_inset
+
+).
+ 
+\end_layout
+
+\begin_layout Standard
+Un anillo es 
+\series bold
+conmutativo
+\series default
+ si su producto es conmutativo, y tiene 
+\series bold
+identidad
+\series default
+ si este tiene elemento neutro 
+\begin_inset Formula $1\in A$
+\end_inset
+
+ llamado 
+\series bold
+uno
+\series default
+.
+ Salvo que se indique lo contrario, al hablar de anillos nos referiremos
+ a anillos conmutativos y con identidad.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ y 
+\begin_inset Formula $\mathbb{Z}_{n}$
+\end_inset
+
+ para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+ son anillos con la suma y el producto usuales.
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $c\in\mathbb{C}$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{Z}[c]\coloneqq\left\{ \sum_{n=0}^{\infty}a_{n}c^{n}\right\} _{a\in\mathbb{Z}^{\mathbb{N}}}\subseteq\mathbb{C}$
+\end_inset
+
+ es un anillo con la suma y el producto de complejos, y en particular lo
+ es 
+\begin_inset Formula $\mathbb{Z}[\text{i}]\coloneqq\{a+b\text{i}\}_{a,b\in\mathbb{Z}}$
+\end_inset
+
+, el 
+\series bold
+anillo de los enteros de Gauss
+\series default
+.
+\end_layout
+
+\begin_layout Enumerate
+El conjunto de funciones 
+\begin_inset Formula $\mathbb{R}\to\mathbb{R}$
+\end_inset
+
+ que se anulan en casi todos los puntos es un anillo conmutativo sin identidad
+ con la suma y producto de funciones.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A_{1},\dots,A_{n}$
+\end_inset
+
+ son anillos, 
+\begin_inset Formula $\prod_{i=1}^{n}A_{i}$
+\end_inset
+
+ es un anillo con las operaciones componente a componente, el 
+\series bold
+anillo producto
+\series default
+ de 
+\begin_inset Formula $A_{1},\dots,A_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Dado un anillo 
+\begin_inset Formula $A\llbracket X\rrbracket\coloneqq A^{\mathbb{N}}$
+\end_inset
+
+ es un anillo con la suma componente a componente y el producto 
+\begin_inset Formula $a\cdot b\coloneqq(\sum_{k=0}^{n}a_{k}b_{n-k})_{n}$
+\end_inset
+
+, el 
+\series bold
+anillo de las series de potencias
+\series default
+ sobre 
+\begin_inset Formula $A$
+\end_inset
+
+, y un 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ se suele escribir con la notación 
+\begin_inset Formula $\sum_{n}a_{n}X^{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El producto tiene precedencia sobre la suma, y escribimos 
+\begin_inset Formula $a-b\coloneqq a+(-b)$
+\end_inset
+
+ y 
+\begin_inset Formula $ab\coloneqq a\cdot b$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un anillo y 
+\begin_inset Formula $a\in A$
+\end_inset
+
+, definimos 
+\begin_inset Formula $0a=0$
+\end_inset
+
+, 
+\begin_inset Formula $a^{0}=1$
+\end_inset
+
+ y, para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $(n+1)a\coloneqq na+a$
+\end_inset
+
+ y 
+\begin_inset Formula $a^{n+1}\coloneqq a^{n}a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dados un anillo 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $a,b,c\in A$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a0=0$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $a0+0=a0=a(0+0)=a0+a0\implies0=a0.$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $-(-a)=a$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $x=-(-a)\implies0=x+(-a)\implies a=x+(-a)+a=x.$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a-b=c\iff b+c=a$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $a-b=c\implies a=a-b+b=c+b=b+c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $b+c=a\implies c=-b+b+c=-b+a=a-b$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(a-b)c=ac-bc$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $(a-b)c+bc=ac-bc+bc=ac\implies ac-bc=(a-b)c.$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(-a)b=-(ab)$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $(-a)b=(0-a)b=0b-ab=0-ab=-ab.$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Anillo trivial si y solo si 
+\begin_inset Formula $1=0$
+\end_inset
+
+, salvo isomorfismo.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dados dos anillos 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+, un 
+\series bold
+homomorfismo de anillos
+\series default
+ es una 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ tal que 
+\begin_inset Formula $f(1)=1$
+\end_inset
+
+ y, para 
+\begin_inset Formula $x,y\in A$
+\end_inset
+
+, 
+\begin_inset Formula $f(x+y)=f(x)+f(y)$
+\end_inset
+
+ y 
+\begin_inset Formula $f(xy)=f(x)f(y)$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $f(0)=0$
+\end_inset
+
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues 
+\begin_inset Formula $f(0)+f(0)=f(0+0)=f(0)=f(0)+0$
+\end_inset
+
+,
+\end_layout
+
+\end_inset
+
+ y 
+\begin_inset Formula $\forall a\in A,f(-a)=-f(a)$
+\end_inset
+
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues 
+\begin_inset Formula $f(-a)+f(a)=f(-a+a)=f(0)=0$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+.
+ Un homomorfismo 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ es inyectivo si y sólo si 
+\begin_inset Formula $\ker f=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Obvio.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $f(a)=f(b)\implies0=f(a)-f(b)=f(a-b)\implies a-b=0\implies a=b$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+isomorfismo de anillos
+\series default
+ es un homomorfismo biyectivo, y entonces su inversa es un homomorfismo.
+ En efecto, sea 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ un isomorfismo, como 
+\begin_inset Formula $f(1)=1$
+\end_inset
+
+, 
+\begin_inset Formula $f^{-1}(1)=1$
+\end_inset
+
+; si 
+\begin_inset Formula $b,b'\in B$
+\end_inset
+
+, sean 
+\begin_inset Formula $a\coloneqq f^{-1}(b)$
+\end_inset
+
+ y 
+\begin_inset Formula $a'\coloneqq f^{-1}(b')$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f(a+a')=f(a)+f(a')=b+b'$
+\end_inset
+
+, luego 
+\begin_inset Formula $f^{-1}(b+b')=a+a'=f^{-1}(b)+f^{-1}(b')$
+\end_inset
+
+, y análogamente 
+\begin_inset Formula $f^{-1}(bb')=f^{-1}(b)f^{-1}(b')$
+\end_inset
+
+.
+ Dos anillos 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+ son 
+\series bold
+isomorfos
+\series default
+, 
+\begin_inset Formula $A\cong B$
+\end_inset
+
+, si existe un isomorfismo entre ellos.
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+anillo cero
+\series default
+ o 
+\series bold
+trivial
+\series default
+ al único con un solo elemento, o en el que 
+\begin_inset Formula $1=0$
+\end_inset
+
+, salvo isomorfismo.
+ En efecto, todo conjunto unipuntual es un anillo con la suma y producto
+ definidos de la única forma posible, la única función entre estos anillos
+ es un isomorfismo y, si el anillo 
+\begin_inset Formula $A$
+\end_inset
+
+ cumple 
+\begin_inset Formula $1=0$
+\end_inset
+
+, para 
+\begin_inset Formula $a\in A$
+\end_inset
+
+, 
+\begin_inset Formula $a=a1=a0=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Elementos notables
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $A$
+\end_inset
+
+ un anillo.
+ Un 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ es 
+\series bold
+invertible
+\series default
+ o 
+\series bold
+unidad
+\series default
+ si existe 
+\begin_inset Formula $b\in A$
+\end_inset
+
+ con 
+\begin_inset Formula $ab=1$
+\end_inset
+
+, en cuyo caso 
+\begin_inset Formula $b$
+\end_inset
+
+ es único, pues 
+\begin_inset Formula $ac=1\implies b=bac=c$
+\end_inset
+
+; lo llamamos 
+\series bold
+inverso
+\series default
+ de 
+\begin_inset Formula $a$
+\end_inset
+
+ o 
+\begin_inset Formula $a^{-1}$
+\end_inset
+
+, y 
+\begin_inset Formula $(a^{-1})^{-1}=a$
+\end_inset
+
+.
+ Llamamos 
+\series bold
+grupo de las unidades
+\series default
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+, 
+\begin_inset Formula $U(A)$
+\end_inset
+
+ o 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+, al grupo abeliano formado por las unidades de 
+\begin_inset Formula $A$
+\end_inset
+
+ con el producto.
+\end_layout
+
+\begin_layout Standard
+Un 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ es 
+\series bold
+cancelable
+\series default
+ si 
+\begin_inset Formula $\forall x,y\in A,(ax=ay\implies x=y)$
+\end_inset
+
+, si y sólo si no es divisor de cero.
+ Toda unidad es cancelable, pues podemos cancelar multiplicando por el inverso.
+ Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es finito se da el recíproco, pues 
+\begin_inset Formula $x\mapsto ax$
+\end_inset
+
+ es inyectiva y por tanto suprayectiva y existe 
+\begin_inset Formula $x$
+\end_inset
+
+ con 
+\begin_inset Formula $ax=1$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $A$
+\end_inset
+
+ infinito esto no es cierto en general, pues 
+\begin_inset Formula $2$
+\end_inset
+
+ es cancelable en 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+ pero no es unidad.
+\end_layout
+
+\begin_layout Standard
+Un 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ es 
+\series bold
+divisor de cero
+\series default
+ si existe 
+\begin_inset Formula $c\in A\setminus\{0\}$
+\end_inset
+
+ con 
+\begin_inset Formula $ac=0$
+\end_inset
+
+, si y sólo si no es cancelable.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Si es cancelable, 
+\begin_inset Formula $ac=0=a0\implies c=0$
+\end_inset
+
+, luego no es divisor de cero.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $x,y\in A$
+\end_inset
+
+ distintos con 
+\begin_inset Formula $ax=ay$
+\end_inset
+
+, entonces 
+\begin_inset Formula $a(x-y)=0$
+\end_inset
+
+, pero 
+\begin_inset Formula $x-y\neq0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Un 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ es 
+\series bold
+nilpotente
+\series default
+ si existe 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+ con 
+\begin_inset Formula $a^{n}=0$
+\end_inset
+
+, en cuyo caso es divisor de 0, pues tomando el menor 
+\begin_inset Formula $n$
+\end_inset
+
+ con 
+\begin_inset Formula $a^{n}=0$
+\end_inset
+
+, 
+\begin_inset Formula $a^{n-1}\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $aa^{n-1}=0$
+\end_inset
+
+.
+ Llamamos 
+\begin_inset Formula $\text{Nil}(A)$
+\end_inset
+
+ al conjunto de elementos de 
+\begin_inset Formula $A$
+\end_inset
+
+ nilpotentes.
+ El 1 es invertible y no nilpotente, y si 
+\begin_inset Formula $A$
+\end_inset
+
+ es no trivial, el 0 es nilpotente y no unidad.
+\end_layout
+
+\begin_layout Standard
+Un anillo es 
+\series bold
+reducido
+\series default
+ si no tiene elementos nilpotentes distintos de 0, si y sólo si todo elemento
+ no nulo tiene cuadrado no nulo.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Trivial.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Si hubiera 
+\begin_inset Formula $b\in\text{Nil}(A)\setminus\{0\}$
+\end_inset
+
+, sea 
+\begin_inset Formula $n>0$
+\end_inset
+
+ mínimo con 
+\begin_inset Formula $b^{n}=0$
+\end_inset
+
+, entonces 
+\begin_inset Formula $b^{n-1}\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $(b^{n-1})^{2}=b^{2n-2}=b^{n}b^{n-2}=0\#$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Un anillo es un 
+\series bold
+dominio
+\series default
+ si no tiene divisores de cero no nulos, si y sólo si todo elemento no nulo
+ es cancelable, y es un 
+\series bold
+cuerpo
+\series default
+ si todo elemento no nulo es unidad.
+ Todo cuerpo es dominio y todo dominio es reducido.
+ Los recíprocos no se cumplen, pues 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+ es un dominio que no es un cuerpo y 
+\begin_inset Formula $\mathbb{Z}_{6}$
+\end_inset
+
+ es un anillo reducido que no es un dominio.
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $n\geq2$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $r\in\mathbb{Z}_{n}$
+\end_inset
+
+ es unidad si y sólo si 
+\begin_inset Formula $\gcd\{r,n\}=1$
+\end_inset
+
+ en 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Si fuera 
+\begin_inset Formula $d\coloneqq\gcd\{r,n\}>1$
+\end_inset
+
+, sean 
+\begin_inset Formula $r',n'\in\mathbb{Z}$
+\end_inset
+
+ con 
+\begin_inset Formula $r=dr'$
+\end_inset
+
+ y 
+\begin_inset Formula $n=dn'$
+\end_inset
+
+, entonces 
+\begin_inset Formula $n'\not\equiv0\bmod n$
+\end_inset
+
+ pero 
+\begin_inset Formula $rn'=dr'n'=r'n\equiv0\bmod n$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $r$
+\end_inset
+
+ es divisor de 0.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Una identidad de Bézout 
+\begin_inset Formula $ar+bn=1$
+\end_inset
+
+ se traduce en que 
+\begin_inset Formula $ar\equiv1\bmod n$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $r\in\mathbb{Z}_{n}$
+\end_inset
+
+ es nilpotente si y sólo si todos los divisores primos de 
+\begin_inset Formula $n$
+\end_inset
+
+ dividen a 
+\begin_inset Formula $r$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $m$
+\end_inset
+
+ con 
+\begin_inset Formula $r^{m}\equiv0$
+\end_inset
+
+ y 
+\begin_inset Formula $p$
+\end_inset
+
+ un divisor primo de 
+\begin_inset Formula $n$
+\end_inset
+
+, como 
+\begin_inset Formula $n$
+\end_inset
+
+ divide a 
+\begin_inset Formula $r^{m}$
+\end_inset
+
+, 
+\begin_inset Formula $p$
+\end_inset
+
+ divide a 
+\begin_inset Formula $r^{m}$
+\end_inset
+
+ y por tanto a 
+\begin_inset Formula $r$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $p_{1}^{k_{1}}\cdots p_{s}^{k_{s}}$
+\end_inset
+
+ la descomposición prima de 
+\begin_inset Formula $n$
+\end_inset
+
+, como 
+\begin_inset Formula $p_{1}\cdots p_{s}$
+\end_inset
+
+ divide a 
+\begin_inset Formula $r$
+\end_inset
+
+, si 
+\begin_inset Formula $m\coloneqq\max\{k_{1},\dots,k_{s}\}$
+\end_inset
+
+, 
+\begin_inset Formula $n$
+\end_inset
+
+ divide a 
+\begin_inset Formula $p_{1}^{m}\cdots p_{s}^{m}$
+\end_inset
+
+ y este a 
+\begin_inset Formula $r$
+\end_inset
+
+, luego 
+\begin_inset Formula $n$
+\end_inset
+
+ divide a 
+\begin_inset Formula $r^{m}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $\mathbb{Z}_{n}$
+\end_inset
+
+ es un cuerpo si y sólo si es un dominio, si y sólo si 
+\begin_inset Formula $n$
+\end_inset
+
+ es primo.
+\end_layout
+
+\begin_deeper
+\begin_layout Description
+\begin_inset Formula $1\implies2]$
+\end_inset
+
+ Obvio.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $2\implies3]$
+\end_inset
+
+ Si 
+\begin_inset Formula $n$
+\end_inset
+
+ no fuera primo, existen 
+\begin_inset Formula $p,q\in\mathbb{Z}$
+\end_inset
+
+, 
+\begin_inset Formula $1<p,q<n$
+\end_inset
+
+, con 
+\begin_inset Formula $n=pq$
+\end_inset
+
+, luego 
+\begin_inset Formula $p$
+\end_inset
+
+ es divisor de 0 en 
+\begin_inset Formula $\mathbb{Z}_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $3\implies1]$
+\end_inset
+
+ Para 
+\begin_inset Formula $r\in\mathbb{Z}_{n}\setminus\{0\}$
+\end_inset
+
+, 
+\begin_inset Formula $\gcd\{r,n\}=1$
+\end_inset
+
+ en 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $r$
+\end_inset
+
+ es unidad.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $\mathbb{Z}_{n}$
+\end_inset
+
+ es reducido si y sólo si 
+\begin_inset Formula $n$
+\end_inset
+
+ es 
+\series bold
+libre de cuadrados
+\series default
+, es decir, si no tiene divisores cuadrados de primos.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Si no fuera libre de cuadrados, sea 
+\begin_inset Formula $n=p^{2}q$
+\end_inset
+
+ para ciertos 
+\begin_inset Formula $p,q\in\mathbb{Z}$
+\end_inset
+
+ con 
+\begin_inset Formula $p$
+\end_inset
+
+ primo, en 
+\begin_inset Formula $\mathbb{Z}_{n}$
+\end_inset
+
+ 
+\begin_inset Formula $pq\neq0$
+\end_inset
+
+ pero 
+\begin_inset Formula $(pq)^{2}=p^{2}q^{2}=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+La descomposición en primos de 
+\begin_inset Formula $n$
+\end_inset
+
+ es de la forma 
+\begin_inset Formula $p_{1}\cdots p_{s}$
+\end_inset
+
+ con los 
+\begin_inset Formula $p_{i}$
+\end_inset
+
+ distintos, y si 
+\begin_inset Formula $r\in\mathbb{Z}_{n}$
+\end_inset
+
+ cumple 
+\begin_inset Formula $r^{2}=0$
+\end_inset
+
+ entonces en 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+ cada 
+\begin_inset Formula $p_{i}$
+\end_inset
+
+ divide a 
+\begin_inset Formula $r^{2}$
+\end_inset
+
+ y por tanto a 
+\begin_inset Formula $r$
+\end_inset
+
+, luego 
+\begin_inset Formula $n$
+\end_inset
+
+ divide a 
+\begin_inset Formula $r$
+\end_inset
+
+ y 
+\begin_inset Formula $r=0$
+\end_inset
+
+ en 
+\begin_inset Formula $\mathbb{Z}_{n}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Section
+Divisibilidad
+\end_layout
+
+\begin_layout Standard
+Dados un dominio 
+\begin_inset Formula $D$
+\end_inset
+
+ y 
+\begin_inset Formula $a,b\in D$
+\end_inset
+
+, 
+\begin_inset Formula $a$
+\end_inset
+
+ 
+\series bold
+divide a
+\series default
+ 
+\begin_inset Formula $b$
+\end_inset
+
+, 
+\begin_inset Formula $a$
+\end_inset
+
+ es 
+\series bold
+divisor
+\series default
+ de 
+\begin_inset Formula $b$
+\end_inset
+
+ o 
+\begin_inset Formula $b$
+\end_inset
+
+ es 
+\series bold
+múltiplo
+\series default
+ de 
+\begin_inset Formula $a$
+\end_inset
+
+, 
+\begin_inset Formula $a\mid b$
+\end_inset
+
+, si existe 
+\begin_inset Formula $c\in D$
+\end_inset
+
+ con 
+\begin_inset Formula $ac=b$
+\end_inset
+
+.
+ Esta relación es reflexiva y transitiva, y para 
+\begin_inset Formula $a,b,c,r,s\in D$
+\end_inset
+
+, si 
+\begin_inset Formula $a\mid b$
+\end_inset
+
+ y 
+\begin_inset Formula $a\mid c$
+\end_inset
+
+, entonces 
+\begin_inset Formula $a\mid rb+sc$
+\end_inset
+
+.
+ Dos elementos 
+\begin_inset Formula $a$
+\end_inset
+
+ y 
+\begin_inset Formula $b$
+\end_inset
+
+ son 
+\series bold
+asociados
+\series default
+ si 
+\begin_inset Formula $a\mid b$
+\end_inset
+
+ y 
+\begin_inset Formula $b\mid a$
+\end_inset
+
+, si y sólo si existe 
+\begin_inset Formula $u\in D^{*}$
+\end_inset
+
+ con 
+\begin_inset Formula $b=au$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $b=0$
+\end_inset
+
+, 
+\begin_inset Formula $a=0$
+\end_inset
+
+ y tomamos 
+\begin_inset Formula $u=1$
+\end_inset
+
+.
+ En otro caso, sean 
+\begin_inset Formula $c,d\in D$
+\end_inset
+
+ con 
+\begin_inset Formula $ac=b$
+\end_inset
+
+ y 
+\begin_inset Formula $bd=a$
+\end_inset
+
+, 
+\begin_inset Formula $b=ac=bdc$
+\end_inset
+
+, luego 
+\begin_inset Formula $dc=1$
+\end_inset
+
+ y 
+\begin_inset Formula $c$
+\end_inset
+
+ es unidad.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $a=bu^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dado un dominio 
+\begin_inset Formula $D$
+\end_inset
+
+, 
+\begin_inset Formula $a\in D\setminus(D^{*}\cup\{0\})$
+\end_inset
+
+ es 
+\series bold
+irreducible
+\series default
+ si para 
+\begin_inset Formula $b,c\in D$
+\end_inset
+
+ con 
+\begin_inset Formula $a=bc$
+\end_inset
+
+, 
+\begin_inset Formula $b\in D^{*}$
+\end_inset
+
+ o 
+\begin_inset Formula $c\in D^{*}$
+\end_inset
+
+.
+ Un 
+\series bold
+dominio de factorización única
+\series default
+ (DFU) es un dominio 
+\begin_inset Formula $D$
+\end_inset
+
+ en el que, para 
+\begin_inset Formula $a\in D\setminus(D^{*}\cup\{0\})$
+\end_inset
+
+, existen 
+\begin_inset Formula $b_{1},\dots,b_{n}\in D$
+\end_inset
+
+ irreducibles con 
+\begin_inset Formula $a=b_{1}\cdots b_{n}$
+\end_inset
+
+, y si 
+\begin_inset Formula $c_{1},\dots,c_{m}\in D$
+\end_inset
+
+ son irreducibles con 
+\begin_inset Formula $a=c_{1}\cdots c_{m}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $n=m$
+\end_inset
+
+ y existe una permutación 
+\begin_inset Formula $\sigma\in{\cal S}_{n}$
+\end_inset
+
+ tal que cada 
+\begin_inset Formula $b_{i}$
+\end_inset
+
+ es asociado con 
+\begin_inset Formula $c_{\sigma(i)}$
+\end_inset
+
+.
+ Todo cuerpo es un DFU, pues no tiene elementos nulos no invertibles.
+ También lo son 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+ y los anillos de polinomios sobre un DFU.
+\end_layout
+
+\begin_layout Section
+Subanillos
+\end_layout
+
+\begin_layout Standard
+Dado un anillo 
+\begin_inset Formula $A$
+\end_inset
+
+, un 
+\begin_inset Formula $S\subseteq A$
+\end_inset
+
+ es un 
+\series bold
+subanillo
+\series default
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+ si es un anillo con las mismas operaciones y el mismo uno que 
+\begin_inset Formula $A$
+\end_inset
+
+, si y sólo si es la imagen de un homomorfismo 
+\begin_inset Formula $B\to A$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $1\in S$
+\end_inset
+
+ y para 
+\begin_inset Formula $x,y\in S$
+\end_inset
+
+, 
+\begin_inset Formula $x-y,xy\in S$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $1\implies2]$
+\end_inset
+
+ Basta tomar el homomorfismo identidad.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $2\implies3]$
+\end_inset
+
+ Sea 
+\begin_inset Formula $f:B\to A$
+\end_inset
+
+ el homomorfismo, 
+\begin_inset Formula $f(1)=1$
+\end_inset
+
+ y, si 
+\begin_inset Formula $x',y'\in B$
+\end_inset
+
+ cumplen 
+\begin_inset Formula $x=f(x')$
+\end_inset
+
+ e 
+\begin_inset Formula $y=f(y')$
+\end_inset
+
+, 
+\begin_inset Formula $x-y=f(x'-y')\in S$
+\end_inset
+
+ y 
+\begin_inset Formula $xy=f(x'y')\in S$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $3\implies1]$
+\end_inset
+
+ 
+\begin_inset Formula $1\in S$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $1-1=0\in S$
+\end_inset
+
+, y para 
+\begin_inset Formula $a,b\in S$
+\end_inset
+
+, 
+\begin_inset Formula $-a=0-a\in S$
+\end_inset
+
+, 
+\begin_inset Formula $a+b=a-(-b)\in S$
+\end_inset
+
+ y 
+\begin_inset Formula $ab\in S$
+\end_inset
+
+, luego 
+\begin_inset Formula $S$
+\end_inset
+
+ es cerrado para suma, producto y opuesto.
+\end_layout
+
+\begin_layout Standard
+Ejemplos:
+\end_layout
+
+\begin_layout Enumerate
+En la cadena 
+\begin_inset Formula $\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}\subseteq\mathbb{C}$
+\end_inset
+
+, cada anillo es subanillo de los que lo contienen, como pasa en 
+\begin_inset Formula $\mathbb{Z}\subseteq\mathbb{Z}[\text{i}]\subseteq\mathbb{C}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Dado un anillo 
+\begin_inset Formula $A$
+\end_inset
+
+, el 
+\series bold
+anillo
+\series default
+ de los polinomios en 
+\begin_inset Formula $A$
+\end_inset
+
+, 
+\begin_inset Formula $A[x]$
+\end_inset
+
+, es el subanillo de 
+\begin_inset Formula $A\llbracket x\rrbracket$
+\end_inset
+
+ formado por las series con una cantidad finita de elementos no nulos, y
+ 
+\begin_inset Formula $A$
+\end_inset
+
+ es un subanillo de 
+\begin_inset Formula $A[x]$
+\end_inset
+
+ identificando 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ con 
+\begin_inset Formula $(a,0,\dots,0,\dots)$
+\end_inset
+
+ por isomorfismo.
+\end_layout
+
+\begin_layout Standard
+Todo subanillo de un dominio es dominio, y todo subanillo de un anillo reducido
+ es reducido.
+ No todo subanillo de un cuerpo es un cuerpo, pues 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+ es subanillo del cuerpo 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ pero no es un cuerpo.
+\end_layout
+
+\begin_layout Section
+Ideales
+\end_layout
+
+\begin_layout Standard
+Un 
+\begin_inset Formula $I\subseteq A$
+\end_inset
+
+ es un 
+\series bold
+ideal
+\series default
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+, 
+\begin_inset Formula $I\trianglelefteq A$
+\end_inset
+
+, si es el núcleo de un homomorfismo 
+\begin_inset Formula $A\to B$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $0\in I$
+\end_inset
+
+ y, para 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $x,y\in I$
+\end_inset
+
+, 
+\begin_inset Formula $x+y,ax\in I$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ un homomorfismo, 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $x,y\in\ker f$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TODO pg.
+ 14, seguir por 11–13, luego por 15.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
-- 
cgit v1.2.3