From ad0ae2bd92011c4002253eb5d15caf82c1f4ad16 Mon Sep 17 00:00:00 2001
From: Juan Marín Noguera <juan.marinn@um.es>
Date: Wed, 13 May 2020 19:44:01 +0200
Subject: Comentadas demostraciones que no entran de GyA

---
 ga/n1.lyx | 506 +++++++++++++++++++++++++++++++++++++++++++++-----------------
 1 file changed, 365 insertions(+), 141 deletions(-)

(limited to 'ga/n1.lyx')

diff --git a/ga/n1.lyx b/ga/n1.lyx
index 8457ada..94f9570 100644
--- a/ga/n1.lyx
+++ b/ga/n1.lyx
@@ -349,10 +349,10 @@ Ejemplos:
 \end_inset
 
  son grupos abelianos.
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 La suma es asociativa y conmutativa con elemento neutro 0, y todo elemento
  
 \begin_inset Formula $a$
@@ -369,7 +369,11 @@ La suma es asociativa y conmutativa con elemento neutro 0, y todo elemento
  solo el 0 tiene opuesto.
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 \begin_inset Formula $\mathbb{N}$
 \end_inset
@@ -391,15 +395,19 @@ La suma es asociativa y conmutativa con elemento neutro 0, y todo elemento
 \end_inset
 
  son monoides conmutativos con el producto.
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 El producto es asociativo y conmutativo con neutro 1, pero el 0 nunca tiene
  opuesto.
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Llamamos 
 \begin_inset Formula $Y^{X}$
@@ -423,10 +431,10 @@ Llamamos
 \end_inset
 
  es un monoide, pero no es conmutativo si hay al menos dos elementos.
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 Claramente 
 \begin_inset Formula $\circ$
 \end_inset
@@ -471,7 +479,11 @@ Claramente
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Llamamos 
 \series bold
@@ -499,14 +511,18 @@ grupo simétrico
 \end_inset
 
  es un grupo.
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 Es asociativa, tiene como neutro la identidad y todo elemento es invertible.
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Sea 
 \begin_inset Formula $X$
@@ -530,10 +546,10 @@ Sea
 
  es un monoide conmutativo cuyos elementos invertibles son las funciones
  que no se anulan.
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 Ambas operaciones son conmutativas y asociativas, la suma tiene como neutro
  la función constante 0 y el producto la función constante 1.
  El inverso de una función 
@@ -563,7 +579,11 @@ Ambas operaciones son conmutativas y asociativas, la suma tiene como neutro
  no se anula.
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 Dada una operación 
 \begin_inset Formula $*$
@@ -605,17 +625,21 @@ Si
 \end_inset
 
  tiene a lo sumo un neutro.
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $f=e*f=e$
 \end_inset
 
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 Dado un monoide 
 \begin_inset Formula $(X,*)$
@@ -655,17 +679,21 @@ Si
 \end_inset
 
  tiene a lo sumo un simétrico.
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $y=e*y=(x*a)*y=x*(a*y)=x*e=x$
 \end_inset
 
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Si 
 \begin_inset Formula $a$
@@ -673,10 +701,10 @@ Si
 
  tiene simétrico por un lado, es cancelable por dicho lado.
  En particular, todo elemento invertible es cancelable.
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 Si, por ejemplo, 
 \begin_inset Formula $a$
 \end_inset
@@ -696,7 +724,11 @@ Si, por ejemplo,
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Section
 Anillos
 \end_layout
@@ -979,10 +1011,10 @@ Todo elemento invertible es regular.
 
 .
  En particular, el 0 y el 1 son únicos.
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $b+a=a\implies b=b+(a-a)=(b+a)-a=a-a=0$
 \end_inset
 
@@ -993,7 +1025,11 @@ Todo elemento invertible es regular.
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 El opuesto de 
 \begin_inset Formula $a$
@@ -1011,10 +1047,10 @@ El opuesto de
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $0a+0a=(0+0)a=0a=0a+0\implies0a=0$
 \end_inset
 
@@ -1025,16 +1061,20 @@ El opuesto de
  se prueba análogamente.
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 \begin_inset Formula $a(-b)=(-a)b=-(ab)$
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $a(-b)+ab=a(-b+b)=a0=0$
 \end_inset
 
@@ -1058,23 +1098,31 @@ El opuesto de
  se prueba análogamente.
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 \begin_inset Formula $a(b-c)=ab-ac$
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $a(b-c)=a(b+(-c))=ab+a(-c)=ab+(-ac)=ab-ac$
 \end_inset
 
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 \begin_inset Formula $a$
 \end_inset
@@ -1096,9 +1144,9 @@ El opuesto de
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
 \begin_layout Enumerate
 \begin_inset Argument item:1
 status open
@@ -1119,6 +1167,7 @@ Basta ver que
 .
 \end_layout
 
+\begin_deeper
 \begin_layout Enumerate
 \begin_inset Argument item:1
 status open
@@ -1169,6 +1218,11 @@ Tenemos
 \end_layout
 
 \end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Si 
 \begin_inset Formula $0=1$
@@ -1179,17 +1233,21 @@ Si
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $a\in A\implies a=a1=a0=0$
 \end_inset
 
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 Dado un anillo 
 \begin_inset Formula $A$
@@ -1278,10 +1336,10 @@ Propiedades: Dados un anillo
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 Para 
 \begin_inset Formula $n=0$
 \end_inset
@@ -1315,16 +1373,20 @@ Para
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 \begin_inset Formula $(n+m)a=na+ma$
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 Para 
 \begin_inset Formula $m=0$
 \end_inset
@@ -1358,16 +1420,20 @@ Para
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 \begin_inset Formula $n(ma)=(nm)a$
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 Para 
 \begin_inset Formula $n=0$
 \end_inset
@@ -1392,7 +1458,11 @@ Para
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Si 
 \begin_inset Formula $n,m\geq0$
@@ -1415,10 +1485,10 @@ Si
 \end_inset
 
  enteros arbitrarios.
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 Para 
 \begin_inset Formula $m=0$
 \end_inset
@@ -1453,7 +1523,7 @@ Para
  
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 Primero vemos que, para 
 \begin_inset Formula $m>0$
 \end_inset
@@ -1502,7 +1572,7 @@ Primero vemos que, para
 .
 \end_layout
 
-\begin_layout Standard
+\begin_layout Plain Layout
 Con esto, sea 
 \begin_inset Formula $m>0$
 \end_inset
@@ -1518,7 +1588,11 @@ Con esto, sea
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Si 
 \begin_inset Formula $A$
@@ -1545,10 +1619,10 @@ Si
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 Para 
 \begin_inset Formula $n=0$
 \end_inset
@@ -1586,7 +1660,11 @@ Para
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Section
 Subanillos
 \end_layout
@@ -1769,7 +1847,8 @@ Para que
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
 \begin_layout Description
 \begin_inset Formula $[1\implies2]$
@@ -1862,6 +1941,11 @@ Para que
 , luego es cerrado para sumas.
 \end_layout
 
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 Algunos subanillos:
 \end_layout
@@ -1915,9 +1999,9 @@ Cada uno de
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
 \begin_layout Enumerate
 \begin_inset Argument item:1
 status open
@@ -1950,6 +2034,7 @@ Si
 .
 \end_layout
 
+\begin_deeper
 \begin_layout Enumerate
 \begin_inset Argument item:1
 status open
@@ -1967,6 +2052,11 @@ Obvio.
 \end_layout
 
 \end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Llamamos 
 \series bold
@@ -1985,10 +2075,10 @@ subanillo primo
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 Claramente 
 \begin_inset Formula $\mathbb{Z}1$
 \end_inset
@@ -2039,7 +2129,11 @@ Claramente
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Si 
 \begin_inset Formula $A$
@@ -2062,14 +2156,18 @@ Si
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 No contiene al 1.
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Dado 
 \begin_inset Formula $z\in\mathbb{C}$
@@ -2277,49 +2375,61 @@ Propiedades: Sean
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $0+f(0)=f(0)=f(0+0)=f(0)+f(0)\implies0=f(0)$
 \end_inset
 
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 \begin_inset Formula $f(-a)=-f(a)$
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $f(a)+f(-a)=f(a+(-a))=f(0)=0$
 \end_inset
 
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 \begin_inset Formula $f(a-b)=f(a)-f(b)$
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $f(a-b)=f(a)+f(-b)=f(a)-f(b)$
 \end_inset
 
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 \begin_inset Formula $f(a_{1}+\dots+a_{n})=f(a_{1})+\dots+f(a_{n})$
 \end_inset
@@ -2332,10 +2442,10 @@ Propiedades: Sean
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 Para 
 \begin_inset Formula $n=0$
 \end_inset
@@ -2347,7 +2457,11 @@ Para
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Si 
 \begin_inset Formula $a$
@@ -2362,10 +2476,10 @@ Si
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $f(a)f(a^{-1})=f(aa^{-1})=f(1)=1$
 \end_inset
 
@@ -2376,7 +2490,11 @@ Si
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 \begin_inset Formula $f(a_{1}\cdots a_{n})=f(a_{1})\cdots f(a_{n})$
 \end_inset
@@ -2402,10 +2520,10 @@ Si
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $1=f(1)\in f(A')$
 \end_inset
 
@@ -2437,7 +2555,11 @@ Si
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Si 
 \begin_inset Formula $B'$
@@ -2456,10 +2578,10 @@ Si
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $1\in f^{-1}(1)\in f^{-1}(B')$
 \end_inset
 
@@ -2491,7 +2613,11 @@ Si
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Si 
 \begin_inset Formula $f$
@@ -2502,10 +2628,10 @@ Si
 \end_inset
 
  también.
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $f^{-1}(1)=1$
 \end_inset
 
@@ -2545,7 +2671,11 @@ Si
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 Ejemplos:
 \end_layout
@@ -2572,9 +2702,9 @@ Dados anillos
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
 \begin_layout Enumerate
 \begin_inset Argument item:1
 status open
@@ -2595,6 +2725,7 @@ status open
 .
 \end_layout
 
+\begin_deeper
 \begin_layout Enumerate
 \begin_inset Argument item:1
 status open
@@ -2624,6 +2755,11 @@ status open
 \end_layout
 
 \end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Sea 
 \begin_inset Formula $B$
@@ -2662,10 +2798,10 @@ Dado un anillo
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $\mu(1)=1$
 \end_inset
 
@@ -2701,7 +2837,11 @@ Dado un anillo
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Dada una familia de anillos 
 \begin_inset Formula $(A_{i})_{i\in I}$
@@ -2810,7 +2950,12 @@ ideal
 \end_inset
 
 .
- Todo ideal contiene al 0, pues tomando 
+ Todo ideal contiene al 0
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues tomando 
 \begin_inset Formula $a\in I$
 \end_inset
 
@@ -2818,6 +2963,11 @@ ideal
 \begin_inset Formula $0=a+(-1)a\in I$
 \end_inset
 
+
+\end_layout
+
+\end_inset
+
 .
  
 \end_layout
@@ -2905,10 +3055,10 @@ ideal principal
 \end_inset
 
  son de esta forma.
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 Sea 
 \begin_inset Formula $I$
 \end_inset
@@ -2984,7 +3134,11 @@ Sea
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Sean 
 \begin_inset Formula $I$
@@ -3092,7 +3246,11 @@ anillo cociente de
 
 \series default
 .
- 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
 \series bold
 Demostración:
 \series default
@@ -3173,6 +3331,11 @@ Demostración:
 .
 \end_layout
 
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 Es claro que 
 \begin_inset Formula $A/0\cong A$
@@ -3192,7 +3355,11 @@ Es claro que
 \end_inset
 
 .
- En efecto, dado 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto, dado 
 \begin_inset Formula $a\in\mathbb{Z}$
 \end_inset
 
@@ -3217,12 +3384,17 @@ Es claro que
 \end_inset
 
 , 
-\begin_inset Formula $a\equiv b\iff a-b\in n\mathbb{Z}\iff n|a-b\overset{|a-b|<n}{\iff}a=b$
+\begin_inset Formula $a\equiv b\iff a-b\in n\mathbb{Z}\iff n\mid a-b\overset{|a-b|<n}{\iff}a=b$
 \end_inset
 
 .
 \end_layout
 
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 Dado un anillo conmutativo 
 \begin_inset Formula $A$
@@ -3240,9 +3412,9 @@ Dado un anillo conmutativo
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
 \begin_layout Enumerate
 \begin_inset Argument item:1
 status open
@@ -3267,6 +3439,7 @@ Dado
 .
 \end_layout
 
+\begin_deeper
 \begin_layout Enumerate
 \begin_inset Argument item:1
 status open
@@ -3296,6 +3469,11 @@ En particular
 \end_layout
 
 \end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Un ideal 
 \begin_inset Formula $I$
@@ -3318,9 +3496,9 @@ Un ideal
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
 \begin_layout Description
 \begin_inset Formula $[1\implies2\implies3]$
 \end_inset
@@ -3328,6 +3506,7 @@ Un ideal
  Obvio.
 \end_layout
 
+\begin_deeper
 \begin_layout Description
 \begin_inset Formula $[3\implies1]$
 \end_inset
@@ -3356,6 +3535,11 @@ Un ideal
 \end_layout
 
 \end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 Sea 
 \begin_inset Formula $f:A\to B$
@@ -3391,7 +3575,11 @@ núcleo
 \end_inset
 
 .
- 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
 \series bold
 Demostración:
 \series default
@@ -3472,6 +3660,11 @@ Demostración:
 .
 \end_layout
 
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 Un homomorfismo de anillos 
 \begin_inset Formula $f:A\to B$
@@ -3482,7 +3675,8 @@ Un homomorfismo de anillos
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
 \begin_layout Itemize
 \begin_inset Argument item:1
@@ -3548,6 +3742,11 @@ Sean
  es inyectiva.
 \end_layout
 
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 
 \series bold
@@ -3947,7 +4146,11 @@ Sean
 \end_inset
 
 .
- En efecto, 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto, 
 \begin_inset Formula $(n)(m)=(\{ab\}_{a\in(n),b\in(m)})=(\{pnqm\}_{p,q\in\mathbb{Z}})=(\{knm\})_{k\in\mathbb{Z}}=(nm)$
 \end_inset
 
@@ -3962,6 +4165,11 @@ Sean
 .
 \end_layout
 
+\end_inset
+
+
+\end_layout
+
 \begin_layout Section
 Teoremas de isomorfía
 \end_layout
@@ -4110,6 +4318,11 @@ Así, si
 \begin_inset Formula $\frac{A\times B}{0\times B}\cong A$
 \end_inset
 
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
 , pues el homomorfismo de proyección 
 \begin_inset Formula $f:A\times B\to A$
 \end_inset
@@ -4122,6 +4335,11 @@ Así, si
 \begin_inset Formula $0\times B$
 \end_inset
 
+
+\end_layout
+
+\end_inset
+
 .
 \end_layout
 
@@ -4421,7 +4639,8 @@ característica
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
 \begin_layout Description
 \begin_inset Formula $[1\implies2]$
@@ -4592,6 +4811,11 @@ característica
 .
 \end_layout
 
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 
 \series bold
-- 
cgit v1.2.3