AC inicio tema 3

2 files changed, 1145 insertions, 0 deletions

diff --git a/ac/n.lyx b/ac/n.lyx
index d405db4..b01fe84 100644
--- a/ac/n.lyx
+++ b/ac/n.lyx
@@ -210,5 +210,19 @@ filename "n2.lyx"
 
 \end_layout
 
+\begin_layout Chapter
+Módulos
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n3.lyx"
+
+\end_inset
+
+
+\end_layout
+
 \end_body
 \end_document
diff --git a/ac/n3.lyx b/ac/n3.lyx
new file mode 100644
index 0000000..6332d29
--- /dev/null
+++ b/ac/n3.lyx
@@ -0,0 +1,1131 @@
+#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
+Dado un anillo 
+\begin_inset Formula $A$
+\end_inset
+
+, un 
+\series bold
+módulo
+\series default
+ sobre 
+\begin_inset Formula $A$
+\end_inset
+
+ o 
+\series bold
+
+\begin_inset Formula $A$
+\end_inset
+
+-módulo
+\series default
+ 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ es una terna 
+\begin_inset Formula $(M,+,\cdot)$
+\end_inset
+
+ donde 
+\begin_inset Formula $(M,+)$
+\end_inset
+
+ es un grupo abeliano y 
+\begin_inset Formula $\cdot:A\times M\to M$
+\end_inset
+
+ es una operación llamada 
+\series bold
+producto por escalares
+\series default
+ tal que para 
+\begin_inset Formula $a,b\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $m,n\in M$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $1m=m$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(ab)m=a(bm)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(a+b)m=am+bm$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a(m+n)=am+an$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $0m=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $1m=(1+0)m=1m+0m$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $a0=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $a0=a(0+0)=a0+a0$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $-(am)=(-a)m=a(-m)$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $am+(-a)m=(a-a)m=0m=0$
+\end_inset
+
+, 
+\begin_inset Formula $am+a(-m)=a(m-m)=a0=0$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Llamamos 
+\begin_inset Formula $_{A}\text{Mod}$
+\end_inset
+
+ a la clase de los módulos sobre 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Dado un cuerpo 
+\begin_inset Formula $K$
+\end_inset
+
+, la clase de espacios vectoriales sobre 
+\begin_inset Formula $K$
+\end_inset
+
+ es 
+\begin_inset Formula $_{K}\text{Vect}=_{K}\text{Mod}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+La clase de grupos abelianos es 
+\begin_inset Formula $\text{GrAb}=_{\mathbb{Z}}\text{Mod}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Un 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+-módulo es un grupo abeliano con un producto por escalares de 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+ y este producto debe cumplir 
+\begin_inset Formula $(a+1)m=am+a$
+\end_inset
+
+ y 
+\begin_inset Formula $(-a)m=a(-m)$
+\end_inset
+
+, por lo que se puede definir de una y sólo una forma.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Llamamos 
+\series bold
+
+\begin_inset Formula $A$
+\end_inset
+
+-módulo regular
+\series default
+ a 
+\begin_inset Formula $_{A}A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Llamamos 
+\series bold
+anulador
+\series default
+ de 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ en 
+\begin_inset Formula $X\subseteq A$
+\end_inset
+
+ a 
+\begin_inset Formula $\text{ann}_{M}(X)\coloneqq\{m\in M:Xm=0\}\leq_{A}M$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\{M_{i}\}_{i\in I}$
+\end_inset
+
+ es una familia de 
+\begin_inset Formula $A$
+\end_inset
+
+-módulos, 
+\begin_inset Formula $\prod_{i\in I}M_{i}$
+\end_inset
+
+ es un 
+\begin_inset Formula $A$
+\end_inset
+
+-módulo con las operaciones componente a componente, y también lo es la
+ 
+\series bold
+suma directa
+\series default
+ (
+\series bold
+externa
+\series default
+) 
+\begin_inset Formula 
+\[
+\bigoplus_{i\in I}M_{i}\coloneqq\left\{ x\in\prod_{i\in I}M_{i}:\{i\in I:x_{i}\neq0\}\text{ finito}\right\} .
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Dados un conjunto 
+\begin_inset Formula $I$
+\end_inset
+
+ y un 
+\begin_inset Formula $A$
+\end_inset
+
+-módulo 
+\begin_inset Formula $M$
+\end_inset
+
+, llamamos 
+\begin_inset Formula $M^{I}\coloneqq\prod_{i\in I}M$
+\end_inset
+
+ y 
+\begin_inset Formula $M^{(I)}\coloneqq\bigoplus_{i\in I}M$
+\end_inset
+
+.
+ Llamamos 
+\series bold
+
+\begin_inset Formula $A$
+\end_inset
+
+-módulo libre de rango 
+\begin_inset Formula $n$
+\end_inset
+
+
+\series default
+ a 
+\begin_inset Formula $_{A}A^{n}$
+\end_inset
+
+, que si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un cuerpo es el espacio vectorial 
+\begin_inset Formula $A^{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Submódulos
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $N\subseteq_{A}M$
+\end_inset
+
+ es un 
+\series bold
+submódulo
+\series default
+ de 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ o un 
+\series bold
+
+\begin_inset Formula $A$
+\end_inset
+
+-submódulo
+\series default
+ de 
+\begin_inset Formula $M$
+\end_inset
+
+, 
+\begin_inset Formula $N\leq_{A}M$
+\end_inset
+
+, si es un subgrupo de 
+\begin_inset Formula $M$
+\end_inset
+
+ cerrado para el producto por escalares, si y sólo si 
+\begin_inset Formula $0\in N$
+\end_inset
+
+ y 
+\begin_inset Formula $N$
+\end_inset
+
+ es cerrado para 
+\series bold
+combinaciones 
+\begin_inset Formula $A$
+\end_inset
+
+-lineales
+\series default
+, en cuyo caso 
+\begin_inset Formula $N$
+\end_inset
+
+ es un módulo.
+\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
+
+
+\begin_inset Formula $N\neq\emptyset$
+\end_inset
+
+ y, para 
+\begin_inset Formula $n\in N$
+\end_inset
+
+, 
+\begin_inset Formula $1n=(1+0)n=1n+0n\implies0n=0\in N$
+\end_inset
+
+, y es claro que es cerrado para combinaciones 
+\begin_inset Formula $A$
+\end_inset
+
+-lineales.
+\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
+
+Claramente es cerrado para la suma y el producto, y también para el opuesto
+ porque si 
+\begin_inset Formula $n\in N$
+\end_inset
+
+ entonces 
+\begin_inset Formula $-n=(-1)n\in N$
+\end_inset
+
+, ya que 
+\begin_inset Formula $n+(-1)n=(1-1)n=0n=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\begin_inset Formula ${\cal L}(_{A}M)$
+\end_inset
+
+ al conjunto de 
+\begin_inset Formula $A$
+\end_inset
+
+-submódulos de 
+\begin_inset Formula $M$
+\end_inset
+
+ ordenado por inclusión, que es un retículo en la que el ínfimo es la intersecci
+ón y el supremo es la suma.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TODO definida más adelante.
+ Añadir demostración.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ejemplos:
+\end_layout
+
+\begin_layout Enumerate
+Dado un cuerpo 
+\begin_inset Formula $K$
+\end_inset
+
+, un 
+\begin_inset Formula $K$
+\end_inset
+
+-submódulo es un subespacio vectorial.
+\end_layout
+
+\begin_layout Enumerate
+Un 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+-submódulo es un subgrupo abeliano.
+\end_layout
+
+\begin_layout Enumerate
+Todo módulo 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ tiene al menos los submódulos 0 y 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+, y puede no haber más.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $_{\mathbb{Z}}\mathbb{Z}_{2}$
+\end_inset
+
+ no tiene más.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Los submódulos del módulo regular son los ideales, 
+\begin_inset Formula ${\cal L}(_{A}A)={\cal L}(A)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es subanillo de 
+\begin_inset Formula $B$
+\end_inset
+
+, 
+\begin_inset Formula $B$
+\end_inset
+
+ es un 
+\begin_inset Formula $A$
+\end_inset
+
+-módulo tomando como producto por escalares el producto en 
+\begin_inset Formula $B$
+\end_inset
+
+.
+ En general 
+\begin_inset Formula ${\cal L}(_{A}B)\neq{\cal L}(_{B}B)$
+\end_inset
+
+.
+ Por ejemplo, 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ solo tiene dos 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+-submódulos (sus ideales) pero tiene muchos 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+-submódulos (sus subgrupos), y dados un anillo 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $\{f\in A[X]:\text{gr}f\leq n\}$
+\end_inset
+
+ es un submódulo de 
+\begin_inset Formula $_{A}A[X]$
+\end_inset
+
+ pero no de 
+\begin_inset Formula $_{A[X]}A[X]$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{ann}_{M}(X)\leq_{A}M$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\bigoplus_{i\in I}M_{i}\leq_{A}\prod_{i\in I}M_{i}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $N\leq_{A}M$
+\end_inset
+
+, 
+\begin_inset Formula $M/N\coloneqq\{\overline{m}\coloneqq m+N\}_{m\in M}$
+\end_inset
+
+ es un 
+\begin_inset Formula $A$
+\end_inset
+
+-módulo con la suma y el producto heredados, el 
+\series bold
+módulo cociente
+\series default
+ de 
+\begin_inset Formula $M$
+\end_inset
+
+ por 
+\begin_inset Formula $N$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Para 
+\begin_inset Formula $m,m',n,n'\in M$
+\end_inset
+
+ y 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ con 
+\begin_inset Formula $m-m',n-n'\in N$
+\end_inset
+
+, 
+\begin_inset Formula $\overline{m}+\overline{n}=\overline{m+n}=\overline{m+n+(m'-m+n'-n)}=\overline{m'+n'}=\overline{m'}+\overline{n'}$
+\end_inset
+
+ y 
+\begin_inset Formula $a\overline{m}=\overline{am}=\overline{am+a(m'-m)}=\overline{am'}=a\overline{m'}$
+\end_inset
+
+, por lo que las operaciones están bien definidas, y es fácil ver que se
+ cumplen los axiomas de 
+\begin_inset Formula $A$
+\end_inset
+
+-módulo.
+\end_layout
+
+\begin_layout Enumerate
+Dado un cuerpo 
+\begin_inset Formula $K$
+\end_inset
+
+, el módulo cociente de 
+\begin_inset Formula $_{K}V$
+\end_inset
+
+ por 
+\begin_inset Formula $U\leq_{K}V$
+\end_inset
+
+ es el espacio vectorial cociente.
+\end_layout
+
+\begin_layout Enumerate
+El módulo cociente de 
+\begin_inset Formula $_{\mathbb{Z}}G$
+\end_inset
+
+ por 
+\begin_inset Formula $H\leq_{\mathbb{Z}}G$
+\end_inset
+
+ es el grupo cociente.
+\end_layout
+
+\begin_layout Section
+Homomorfismos
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+homomorfismo de 
+\begin_inset Formula $A$
+\end_inset
+
+-módulos
+\series default
+, 
+\series bold
+
+\begin_inset Formula $A$
+\end_inset
+
+-homomorfismo
+\series default
+ o 
+\series bold
+aplicación 
+\begin_inset Formula $A$
+\end_inset
+
+-lineal
+\series default
+ entre 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ y 
+\begin_inset Formula $_{A}N$
+\end_inset
+
+ es un homomorfismo de grupos abelianos 
+\begin_inset Formula $f:M\to N$
+\end_inset
+
+ que conserva el producto por escalares, y llamamos 
+\begin_inset Formula $\text{Hom}_{A}(M,N)$
+\end_inset
+
+ al conjunto de los 
+\begin_inset Formula $A$
+\end_inset
+
+-homomorfismos de 
+\begin_inset Formula $M$
+\end_inset
+
+ a 
+\begin_inset Formula $N$
+\end_inset
+
+, que es un grupo abeliano con la suma.
+ El 
+\series bold
+núcleo
+\series default
+ de un 
+\begin_inset Formula $A$
+\end_inset
+
+-homomorfismo 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\begin_inset Formula $\ker f\coloneqq f^{-1}(0)$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $f$
+\end_inset
+
+ es inyectivo si y sólo si 
+\begin_inset Formula $\ker f=0$
+\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
+
+
+\begin_inset Formula $f(x)=0=f(0)\implies x=0$
+\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
+
+
+\begin_inset Formula $f(a)=f(b)\implies f(a-b)=0\implies a-b=0$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $M'\leq_{A}M$
+\end_inset
+
+, 
+\begin_inset Formula $f(M')\leq_{A}N$
+\end_inset
+
+, y en particular 
+\begin_inset Formula $f(M)\leq_{A}N$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $f(M')$
+\end_inset
+
+ contiene al 
+\begin_inset Formula $0=f(0)$
+\end_inset
+
+ y sus combinaciones 
+\begin_inset Formula $A$
+\end_inset
+
+-lineales son la imagen de combinaciones 
+\begin_inset Formula $A$
+\end_inset
+
+-lineales en 
+\begin_inset Formula $M'$
+\end_inset
+
+, que están en 
+\begin_inset Formula $M'$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $N'\leq_{A}N$
+\end_inset
+
+, 
+\begin_inset Formula $f^{-1}(N')\leq_{A}M$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $f^{-1}(N')$
+\end_inset
+
+ contiene al 
+\begin_inset Formula $0=f^{-1}(N')$
+\end_inset
+
+, y si 
+\begin_inset Formula $a_{1},\dots,a_{k}\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $m_{1},\dots,m_{k}\in f^{-1}(N')$
+\end_inset
+
+, 
+\begin_inset Formula $f(a_{1}m_{1}+\dots+a_{k}m_{k})=a_{1}f(m_{1})+\dots+a_{k}f(m_{k})\in N'$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+La composición de 
+\begin_inset Formula $A$
+\end_inset
+
+-homomorfismos es un 
+\begin_inset Formula $A$
+\end_inset
+
+-homomorfismos.
+\end_layout
+
+\begin_layout Standard
+Un homomorfismo es un 
+\series bold
+monomorfismo
+\series default
+ si es inyectivo, un 
+\series bold
+epimorfismo
+\series default
+ si es suprayectivo y un 
+\series bold
+isomorfismo
+\series default
+ si es biyectivo.
+ Las proyecciones canónicas 
+\begin_inset Formula $M\to M/N$
+\end_inset
+
+ son epimorfismos.
+ Los inversos de isomorfismos son isomorfismos, y se dice que los módulos
+ involucrados son 
+\series bold
+isomorfos
+\series default
+.
+ En efecto, si 
+\begin_inset Formula $f:M\to N$
+\end_inset
+
+ es un 
+\begin_inset Formula $A$
+\end_inset
+
+-isomorfismo, 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $n,n'\in N$
+\end_inset
+
+ con imágenes por 
+\begin_inset Formula $f^{-1}$
+\end_inset
+
+ 
+\begin_inset Formula $m,m'\in M$
+\end_inset
+
+, 
+\begin_inset Formula $f(m+m')=f(m)+f(m')=n+n'$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $f^{-1}(n+n')=m+m'=f^{-1}(n)+f^{-1}(n')$
+\end_inset
+
+ y 
+\begin_inset Formula $f(am)=af(m)=an$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $f^{-1}(an)=am=af^{-1}(n)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+En un cuerpo 
+\begin_inset Formula $K$
+\end_inset
+
+, un 
+\begin_inset Formula $K$
+\end_inset
+
+-homomorfismo es una aplicación lineal.
+\end_layout
+
+\begin_layout Enumerate
+Un 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+-homomorfismo es un homomorfismo de grupos.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{Hom}_{\mathbb{Z}}(\mathbb{Z}_{2},\mathbb{Z}_{3})=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TODO 4.1.8
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Restricción de escalares
+\end_layout
+
+\begin_layout Standard
+Dado un homomorfismo de anillos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+, cada 
+\begin_inset Formula $B$
+\end_inset
+
+-módulo es un 
+\begin_inset Formula $A$
+\end_inset
+
+-módulo por 
+\series bold
+restricción de escalares
+\series default
+, 
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TODO
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document