Terminados apuntes de Álgebra Conmutativa

1 files changed, 1439 insertions, 3 deletions

diff --git a/ac/n3.lyx b/ac/n3.lyx
index 4da49b6..9f5500e 100644
--- a/ac/n3.lyx
+++ b/ac/n3.lyx
@@ -161,6 +161,55 @@ producto por escalares
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Equivalentemente, el producto currificado es un homomorfismo de anillos
+ 
+\begin_inset Formula $A\to\text{End}(M)$
+\end_inset
+
+, donde 
+\begin_inset Formula $\text{End}(M)$
+\end_inset
+
+ es el anillo de los endomorfismos del grupo abeliano 
+\begin_inset Formula $M$
+\end_inset
+
+ con la suma por componentes y la composición como producto.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Propiedades:
 \end_layout
 
@@ -304,7 +353,7 @@ anulador
 \end_inset
 
  a 
-\begin_inset Formula $\text{ann}_{M}(X)\coloneqq\{m\in M\mid Xm=0\}\leq_{A}M$
+\begin_inset Formula $\text{ann}_{M}(X)\coloneqq\{m\in M\mid Xm=0\}$
 \end_inset
 
 .
@@ -665,6 +714,10 @@ Si
 \begin_inset Formula $\text{ann}_{M}(X)\leq_{A}M$
 \end_inset
 
+, y en particular 
+\begin_inset Formula $\text{ann}_{A}(X)\trianglelefteq A$
+\end_inset
+
 .
 \end_layout
 
@@ -676,6 +729,104 @@ Si
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+8.
+\end_layout
+
+\end_inset
+
+Para 
+\begin_inset Formula $I\trianglelefteq A$
+\end_inset
+
+ y 
+\begin_inset Formula $X\subseteq_{A}M$
+\end_inset
+
+, 
+\begin_inset Formula $IX\leq_{A}M$
+\end_inset
+
+, y en particular, para 
+\begin_inset Formula $m\in M$
+\end_inset
+
+, 
+\begin_inset Formula $Im=\{bm\}_{b\in I}\leq_{A}M$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+9.
+\end_layout
+
+\end_inset
+
+Para 
+\begin_inset Formula $S\subseteq A$
+\end_inset
+
+ y 
+\begin_inset Formula $N\leq_{A}M$
+\end_inset
+
+, 
+\begin_inset Formula $SN\leq_{A}M$
+\end_inset
+
+, y en particular, para 
+\begin_inset Formula $a\in A$
+\end_inset
+
+, 
+\begin_inset Formula $aN=\{an\}_{n\in N}\leq_{A}M$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Si 
 \begin_inset Formula $N\leq_{A}M$
 \end_inset
@@ -1109,6 +1260,103 @@ Un
 .
 \end_layout
 
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Que dos submódulos de 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ sean isomorfos no significa que lo sean los módulos cociente de 
+\begin_inset Formula $M$
+\end_inset
+
+ entre ellos, ni al revés.
+ Por ejemplo, si 
+\begin_inset Formula $_{\mathbb{Z}}M\coloneqq\mathbb{Z}_{3}\oplus\mathbb{Z}_{9}$
+\end_inset
+
+, 
+\begin_inset Formula $K\coloneqq\mathbb{Z}_{3}\oplus0$
+\end_inset
+
+, 
+\begin_inset Formula $N\coloneqq0\oplus\mathbb{Z}_{9}$
+\end_inset
+
+ y 
+\begin_inset Formula $L=((0,6))$
+\end_inset
+
+, 
+\begin_inset Formula $K\cong L$
+\end_inset
+
+ pero 
+\begin_inset Formula $\frac{M}{K}\ncong\frac{M}{L}$
+\end_inset
+
+, y 
+\begin_inset Formula $\frac{M}{K+L}\cong\frac{M}{N}$
+\end_inset
+
+ pero 
+\begin_inset Formula $K+L\ncong N$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\phi:M\to M'$
+\end_inset
+
+ es un 
+\begin_inset Formula $A$
+\end_inset
+
+-isomorfismo, para 
+\begin_inset Formula $N\leq_{A}M$
+\end_inset
+
+, 
+\begin_inset Formula $\frac{M}{N}\cong\frac{M'}{\phi(N)}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Section
 Restricción de escalares
 \end_layout
@@ -1616,6 +1864,94 @@ Finalmente, estas operaciones son inversas una de la otra, pues para
 \end_layout
 
 \end_deeper
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $V$
+\end_inset
+
+ y 
+\begin_inset Formula $W$
+\end_inset
+
+ 
+\begin_inset Formula $K$
+\end_inset
+
+-espacios vectoriales y 
+\begin_inset Formula $f:V\to V$
+\end_inset
+
+ y 
+\begin_inset Formula $g:V\to V$
+\end_inset
+
+ 
+\begin_inset Formula $K$
+\end_inset
+
+-endomorfismos, un 
+\begin_inset Formula $K[X]$
+\end_inset
+
+-homomorfismo entre los 
+\begin_inset Formula $K[X]$
+\end_inset
+
+-módulos asociados a 
+\begin_inset Formula $(V,f)$
+\end_inset
+
+ y 
+\begin_inset Formula $(W,g)$
+\end_inset
+
+ es precisamente una aplicación 
+\begin_inset Formula $K$
+\end_inset
+
+-lineal 
+\begin_inset Formula $\phi:V\to W$
+\end_inset
+
+ con 
+\begin_inset Formula $\phi\circ f=g\circ\phi$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Section
 Teoremas de isomorfía
 \end_layout
@@ -1928,8 +2264,73 @@ Sea
 \end_layout
 
 \end_deeper
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+clase de isomorfía
+\series default
+ es una clase de equivalencia por la relación 
+\begin_inset Quotes cld
+\end_inset
+
+ser isomorfos
+\begin_inset Quotes crd
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $I,J\trianglelefteq A$
+\end_inset
+
+, si 
+\begin_inset Formula $\frac{A}{I}\cong\frac{A}{J}$
+\end_inset
+
+ como 
+\begin_inset Formula $A$
+\end_inset
+
+-módulos entonces 
+\begin_inset Formula $I=J$
+\end_inset
+
+, pero esto no es válido si el isomorfismo es de anillos.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Section
-Operaciones con submódulos
+Sistemas generadores
 \end_layout
 
 \begin_layout Standard
@@ -2291,6 +2692,265 @@ Si
 
 \end_deeper
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+9.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $N\leq_{A}M$
+\end_inset
+
+ y 
+\begin_inset Formula $\frac{M}{N}$
+\end_inset
+
+ son finitamente generados, 
+\begin_inset Formula $M$
+\end_inset
+
+ es finitamente generado.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+10.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $N,K\leq_{A}M$
+\end_inset
+
+, 
+\begin_inset Formula $N\cap K\eqqcolon(x_{1},\dots,x_{r})$
+\end_inset
+
+, 
+\begin_inset Formula $N+K\eqqcolon(y_{1},\dots,y_{s})$
+\end_inset
+
+ y, para 
+\begin_inset Formula $j\in\{1,\dots,s\}$
+\end_inset
+
+, 
+\begin_inset Formula $y_{j}\eqqcolon n_{j}+k_{j}$
+\end_inset
+
+ con 
+\begin_inset Formula $n_{j}\in N$
+\end_inset
+
+ y 
+\begin_inset Formula $k_{j}\in K$
+\end_inset
+
+, entonces 
+\begin_inset Formula $N=(x_{1},\dots,x_{r},n_{1},\dots,n_{s})$
+\end_inset
+
+ y 
+\begin_inset Formula $K=(x_{1},\dots,x_{r},k_{1},\dots,k_{s})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+11.
+\end_layout
+
+\end_inset
+
+Dado un entero 
+\begin_inset Formula $q\geq2$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{Z}\left[\frac{1}{q}\right]=\left\{ \frac{a}{q^{n}}\right\} _{a\in\mathbb{Z},n\in\mathbb{N}}\leq_{\mathbb{Z}}\mathbb{Q}$
+\end_inset
+
+ no es finitamente generado.
+ 
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+12.
+\end_layout
+
+\end_inset
+
+Los epimorfismos conservan los conjuntos generadores.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Lema de Nakayama:
+\series default
+ Dados 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ y 
+\begin_inset Formula $J\leq A$
+\end_inset
+
+ con 
+\begin_inset Formula $J\subseteq\text{Jac}A$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $M$
+\end_inset
+
+ es finitamente generado y 
+\begin_inset Formula $JM=M$
+\end_inset
+
+ entonces 
+\begin_inset Formula $M=0$
+\end_inset
+
+.
+ Esto no se cumple si 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ no es finitamente generado, pues por ejemplo 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ visto como 
+\begin_inset Formula $\mathbb{Z}_{(p)}$
+\end_inset
+
+-módulo cumple 
+\begin_inset Formula $\text{Jac}(\mathbb{Z}_{p}(\mathbb{Q}))=\mathbb{Q}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $M$
+\end_inset
+
+ es finitamente generado, el único 
+\begin_inset Formula $N\leq_{A}M$
+\end_inset
+
+ con 
+\begin_inset Formula $M=JM+N$
+\end_inset
+
+ es 
+\begin_inset Formula $M$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $(A,J,K)$
+\end_inset
+
+ es un anillo local, 
+\begin_inset Formula $\frac{M}{JM}$
+\end_inset
+
+ es anulado por 
+\begin_inset Formula $J$
+\end_inset
+
+ (
+\begin_inset Formula $J\subseteq\text{ann}_{A}(\frac{M}{JM})$
+\end_inset
+
+), luego es un 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial.
+ Si además 
+\begin_inset Formula $M$
+\end_inset
+
+ es finitamente generado, 
+\begin_inset Formula $\frac{M}{JM}$
+\end_inset
+
+ es de dimensión finita, y si 
+\begin_inset Formula $_{K}\frac{M}{JM}=(\overline{m_{1}},\dots,\overline{m_{n}})$
+\end_inset
+
+ entonces 
+\begin_inset Formula $_{A}M=(m_{1},\dots,m_{n})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Sumas directas
+\end_layout
+
+\begin_layout Standard
 Sean 
 \begin_inset Formula $\{N_{i}\}_{i\in I}\subseteq{\cal L}(_{A}M)$
 \end_inset
@@ -2571,6 +3231,100 @@ La unión de un conjunto generador de
 
 \end_deeper
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $J\trianglelefteq A$
+\end_inset
+
+ y 
+\begin_inset Formula $_{A}M=\bigoplus_{i\in I}M_{i}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Dado un 
+\begin_inset Formula $A$
+\end_inset
+
+-isomorfismo 
+\begin_inset Formula $\phi:M\to N$
+\end_inset
+
+, 
+\begin_inset Formula $N=\bigoplus_{i\in I}f(M_{i})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{ann}_{M}(J)=\bigoplus_{i\in I}\text{ann}_{M_{i}}(J)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{ann}_{A}(M)=\bigcap_{i\in I}\text{ann}_{A}(M_{i})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un DIP, 
+\begin_inset Formula $I$
+\end_inset
+
+ es finito y 
+\begin_inset Formula $\text{ann}_{A}(M_{i})=(b_{i})$
+\end_inset
+
+ para cada 
+\begin_inset Formula $i\in I$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\text{ann}_{A}(M)=(\text{lcm}_{i\in I}b_{i})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 \begin_inset Formula $N\leq_{A}M$
 \end_inset
 
@@ -3282,6 +4036,88 @@ TODO ejercicio Saorín 2
 
 \end_layout
 
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+8.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $e\in A$
+\end_inset
+
+ es idempotente, 
+\begin_inset Formula $eM$
+\end_inset
+
+ es sumando directo de 
+\begin_inset Formula $M$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+9.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $f:M\to M$
+\end_inset
+
+ es un 
+\begin_inset Formula $A$
+\end_inset
+
+-endomorfismo idempotente, 
+\begin_inset Formula $M=\ker f\oplus\text{Im}f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Section
 Módulos libres
 \end_layout
@@ -3600,7 +4436,78 @@ begin{exinfo}
 
 \end_inset
 
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+8.
+\end_layout
+
+\end_inset
+
+Los epimorfismos conservan la independencia lineal.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+9.
+\end_layout
+
+\end_inset
+
 Los isomorfismos conservan bases.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+10.
+\end_layout
+
+\end_inset
+
+Un 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+-submódulo de 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ es libre si y sólo si es cíclico, si y solo si es finitamente generado.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+11.
+\end_layout
+
+\end_inset
+
+Un anillo 
+\begin_inset Formula $A$
+\end_inset
+
+ es un cuerpo si y sólo si todo 
+\begin_inset Formula $A$
+\end_inset
+
+-módulo es libre.
+\end_layout
+
+\begin_layout Standard
 \begin_inset ERT
 status open
 
@@ -4209,6 +5116,43 @@ Sean
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $_{A}N\in{\cal L}(_{A}M)$
+\end_inset
+
+ es 
+\series bold
+finitamente cogenerado
+\series default
+ si es cocompacto.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 \begin_inset Formula $_{A}M$
 \end_inset
 
@@ -4497,6 +5441,105 @@ Como todos sus subgrupos son los de esta cadena,
 
 \end_deeper
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+5.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\frac{\mathbb{Q}}{\mathbb{Z}}=\bigoplus_{p}\mathbb{Z}_{p^{\infty}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+6.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ es noetheriano, todo 
+\begin_inset Formula $A$
+\end_inset
+
+-endomorfismo suprayectivo en 
+\begin_inset Formula $M$
+\end_inset
+
+ es inyectivo.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+7.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ es artiniano, todo 
+\begin_inset Formula $A$
+\end_inset
+
+-endomorfismo inyectivo en 
+\begin_inset Formula $M$
+\end_inset
+
+ es suprayectivo.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Una 
 \series bold
 sucesión exacta corta
@@ -5032,7 +6075,7 @@ Si
 \end_inset
 
 -módulo, y en particular 
-\begin_inset Formula ${\cal L}(_{A}M)\cong{\cal L}(_{A_{1}}M_{1})\times\dots\times{\cal L}(_{A_{n}}M_{n})$
+\begin_inset Formula ${\cal L}(_{A}M)\cong\prod_{i=1}^{m}{\cal L}(_{A_{i}}M_{i})$
 \end_inset
 
 .
@@ -5306,5 +6349,398 @@ de longitud finita
 -módulo finitamente generado es de longitud finita.
 \end_layout
 
+\begin_layout Section
+Módulos y matrices
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $m,n\in\mathbb{N}^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal C}_{m}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal C}_{n}$
+\end_inset
+
+ las bases canónicas respectivas de los 
+\begin_inset Formula $A$
+\end_inset
+
+-módulos libres 
+\begin_inset Formula $A^{m}$
+\end_inset
+
+ y 
+\begin_inset Formula $A^{n}$
+\end_inset
+
+, 
+\begin_inset Formula $(f\mapsto M_{{\cal C}_{m}{\cal C}_{n}}(f)):\text{Hom}_{A}(A^{n},A^{m})\to{\cal M}_{m\times n}(A)$
+\end_inset
+
+ es un isomorfismo de 
+\begin_inset Formula $A$
+\end_inset
+
+-módulos con inversa 
+\begin_inset Formula $C\mapsto v\mapsto Cv$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula ${\cal C}_{n}\eqqcolon(e_{1},\dots,e_{n})$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal C}_{m}\eqqcolon(f_{1},\dots,f_{m})$
+\end_inset
+
+, toda 
+\begin_inset Formula $f\in\text{Hom}_{A}(A^{n},A^{m})$
+\end_inset
+
+ viene dada por los valores que le asigna a los 
+\begin_inset Formula $e_{i}$
+\end_inset
+
+, que se pueden expresar respecto a los 
+\begin_inset Formula $f_{j}$
+\end_inset
+
+ dando lugar a 
+\begin_inset Formula $M\coloneqq M_{{\cal C}_{m}{\cal C}_{n}}(f)$
+\end_inset
+
+ cuyas columnas son los 
+\begin_inset Formula $f(e_{i})$
+\end_inset
+
+, pero claramente 
+\begin_inset Formula $Me_{i}$
+\end_inset
+
+ es la 
+\begin_inset Formula $i$
+\end_inset
+
+-ésima columna de 
+\begin_inset Formula $M$
+\end_inset
+
+, y recíprocamente, si 
+\begin_inset Formula $M\in{\cal M}_{m\times n}(A)$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ viene dada por 
+\begin_inset Formula $f(v)\coloneqq Mv$
+\end_inset
+
+, las columnas de 
+\begin_inset Formula $M_{{\cal C}_{m}{\cal C}_{n}}(f)$
+\end_inset
+
+ son los 
+\begin_inset Formula $Me_{i}$
+\end_inset
+
+ que son las columnas de 
+\begin_inset Formula $M$
+\end_inset
+
+.
+ Que es un isomorfismo es claro tomando 
+\begin_inset Formula $(b_{ij}\coloneqq\sum_{k}a_{k}e_{k}\mapsto a_{i}f_{j})_{i,j}$
+\end_inset
+
+ como base de 
+\begin_inset Formula $\text{Hom}_{A}(A^{n},A^{m})$
+\end_inset
+
+ y viendo que conserva combinaciones lineales de los 
+\begin_inset Formula $b_{ij}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\text{GL}_{s}(K)\coloneqq\{A\in{\cal M}_{s}(K)\mid\det A\neq0\}$
+\end_inset
+
+.
+ Dada 
+\begin_inset Formula $C\in{\cal M}_{m\times n}(A)$
+\end_inset
+
+, llamamos 
+\series bold
+
+\begin_inset Formula $A$
+\end_inset
+
+-módulo asociado a 
+\begin_inset Formula $C$
+\end_inset
+
+
+\series default
+, 
+\begin_inset Formula $M(C)$
+\end_inset
+
+, a 
+\begin_inset Formula $\frac{A^{m}}{\{Cv\}_{v\in A^{n}}}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $B,C\in{\cal M}_{m\times n}(A)$
+\end_inset
+
+ son 
+\series bold
+equivalentes
+\series default
+ si existen 
+\begin_inset Formula $P\in\text{GL}_{m}(A)$
+\end_inset
+
+ y 
+\begin_inset Formula $Q\in\text{GL}_{n}(A)$
+\end_inset
+
+ con 
+\begin_inset Formula $C=PBQ$
+\end_inset
+
+, en cuyo caso 
+\begin_inset Formula $M(B)\cong M(C)$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Se tiene 
+\begin_inset Formula $PB=CQ^{-1}$
+\end_inset
+
+, luego llamando 
+\begin_inset Formula $f_{C}:A^{n}\to A^{m}$
+\end_inset
+
+ al homomorfismo 
+\begin_inset Formula $f_{C}(v)\coloneqq Cv$
+\end_inset
+
+, 
+\begin_inset Formula $f_{P}\circ f_{B}=f_{C}\circ f_{Q^{-1}}$
+\end_inset
+
+.
+ Definiendo el homomorfismo 
+\begin_inset Formula $\psi:M(B)\to M(C)$
+\end_inset
+
+ como 
+\begin_inset Formula $\psi(\overline{a})=\overline{f_{P}(a)}$
+\end_inset
+
+, 
+\begin_inset Formula $\psi$
+\end_inset
+
+ está bien definido porque 
+\begin_inset Formula $a\in\text{Im}f_{B}\implies f_{P}(a)\in\text{Im}(f_{P}\circ f_{B})=\text{Im}(f_{C}\circ f_{Q^{-1}})=\text{Im}f_{C}$
+\end_inset
+
+, pero el homomorfismo 
+\begin_inset Formula $\phi:M(C)\to M(B)$
+\end_inset
+
+ dado por 
+\begin_inset Formula $\phi(\overline{c})\coloneqq\overline{f_{P^{-1}}(c)}$
+\end_inset
+
+ también está bien definido porque 
+\begin_inset Formula $c\in\text{Im}f_{C}\implies f_{P^{-1}}(c)\in\text{Im}(f_{P^{-1}}\circ f_{C})=\text{Im}(f_{P^{-1}}\circ f_{C}\circ f_{Q^{-1}})=\text{Im}(f_{P})$
+\end_inset
+
+, y 
+\begin_inset Formula $\phi=\psi^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+operación
+\series default
+ o 
+\series bold
+transformación elemental por filas
+\series default
+ o 
+\series bold
+columnas
+\series default
+ en 
+\begin_inset Formula $C\in{\cal M}_{m\times n}(A)$
+\end_inset
+
+ consiste en intercambiar dos filas o columnas de 
+\begin_inset Formula $C$
+\end_inset
+
+, multiplicar una por un 
+\begin_inset Formula $\alpha\in A^{*}$
+\end_inset
+
+ o sumarle a una otra multiplicada por un 
+\begin_inset Formula $\alpha\in A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{AlgL}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+matriz elemental
+\series default
+ de tamaño 
+\begin_inset Formula $n$
+\end_inset
+
+ a toda matriz obtenida al efectuar una operación elemental [...] en 
+\begin_inset Formula $I_{n}$
+\end_inset
+
+.
+ [...] Si 
+\begin_inset Formula $B$
+\end_inset
+
+ se obtiene al realizar una operación elemental por filas en 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $E$
+\end_inset
+
+ al realizar la misma en 
+\begin_inset Formula $I_{m}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $B=EA$
+\end_inset
+
+.
+ [...] Si 
+\begin_inset Formula $B$
+\end_inset
+
+ se obtiene de aplicar una operación elemental por columnas en 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $E$
+\end_inset
+
+ al aplicarla a 
+\begin_inset Formula $I_{n}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $B=AE$
+\end_inset
+
+.
+ Así, realizar una serie de estas operaciones en una matriz equivale a multiplic
+arla por uno o ambos lados por un producto de matrices elementales, el cual
+ es invertible.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Las matrices elementales son las mismas por filas que por columnas.
+ Si 
+\begin_inset Formula $B,C\in{\cal M}_{m\times n}(A)$
+\end_inset
+
+ y 
+\begin_inset Formula $C$
+\end_inset
+
+ se puede obtener aplicando a 
+\begin_inset Formula $B$
+\end_inset
+
+ una cantidad finita de transformaciones elementales por filas y por columnas,
+ entonces 
+\begin_inset Formula $B$
+\end_inset
+
+ y 
+\begin_inset Formula $C$
+\end_inset
+
+ son equivalentes, pues aplicar transformaciones por filas y columnas a
+ 
+\begin_inset Formula $B$
+\end_inset
+
+ equivale a multiplicarla a izquierda y derecha por matrices invertibles.
+\end_layout
+
 \end_body
 \end_document