Ejercicios tema 1

1 files changed, 992 insertions, 17 deletions

diff --git a/ac/n1.lyx b/ac/n1.lyx
index ca8140c..1d91f9d 100644
--- a/ac/n1.lyx
+++ b/ac/n1.lyx
@@ -723,6 +723,15 @@ grupo de las unidades
 \end_inset
 
  con el producto.
+ Para 
+\begin_inset Formula $x,y\in A$
+\end_inset
+
+, 
+\begin_inset Formula $xy\in A^{*}\iff x,y\in A^{*}$
+\end_inset
+
+.
 \end_layout
 
 \begin_layout Standard
@@ -903,6 +912,183 @@ nilradical
 \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
+Si 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ es nilpotente entonces 
+\begin_inset Formula $1+(a)\subseteq A^{*}$
+\end_inset
+
+ y, para 
+\begin_inset Formula $u\in A^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $u+a\in U(A)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Un 
+\begin_inset Formula $e\in A$
+\end_inset
+
+ es 
+\series bold
+idempotente
+\series default
+ si 
+\begin_inset Formula $e^{2}=e$
+\end_inset
+
+, en cuyo caso 
+\begin_inset Formula $f\coloneqq1-e$
+\end_inset
+
+ también lo es y 
+\begin_inset Formula $ef=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dados anillos 
+\begin_inset Formula $A_{1},\dots,A_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $a=(a_{1},\dots,a_{n})\in A\coloneqq A_{1}\times\dots\times A_{n}$
+\end_inset
+
+, 
+\begin_inset Formula $a$
+\end_inset
+
+ es invertible, cancelable, divisor de cero, nilpotente o idempotente en
+ 
+\begin_inset Formula $A$
+\end_inset
+
+ si y sólo si lo es cada 
+\begin_inset Formula $a_{i}$
+\end_inset
+
+ en 
+\begin_inset Formula $A_{i}$
+\end_inset
+
+.
+\begin_inset Quotes crd
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $m\in\mathbb{Z}$
+\end_inset
+
+ no cuadrado, definimos la 
+\series bold
+norma
+\series default
+ en 
+\begin_inset Formula $\mathbb{Z}[\sqrt{m}]$
+\end_inset
+
+ como 
+\begin_inset Formula $N:\mathbb{Z}[\sqrt{m}]\to\mathbb{Z}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $N(a+b\sqrt{m})\coloneqq a^{2}-mb^{2}$
+\end_inset
+
+ para 
+\begin_inset Formula $a,b\in\mathbb{Z}$
+\end_inset
+
+, y entonces:
+\end_layout
+
+\begin_layout Enumerate
+Las unidades de 
+\begin_inset Formula $\mathbb{Z}[\sqrt{m}]$
+\end_inset
+
+ son los elementos de norma 1.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $m<0$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\mathbb{Z}[\sqrt{m}]^{*}$
+\end_inset
+
+ es finito.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $m>0$
+\end_inset
+
+ y 
+\begin_inset Formula $|\mathbb{Z}[\sqrt{m}]|^{*}>2$
+\end_inset
+
+ entonces 
+\begin_inset Formula $|\mathbb{Z}[\sqrt{m}]|^{*}=|\mathbb{N}|$
+\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
+Dominios
+\end_layout
+
+\begin_layout Standard
 Un anillo es 
 \series bold
 reducido
@@ -987,13 +1173,6 @@ cuerpo
 \end_layout
 
 \begin_layout Standard
-\begin_inset Newpage pagebreak
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
 Para 
 \begin_inset Formula $n\geq2$
 \end_inset
@@ -1410,8 +1589,34 @@ La descomposición en primos de
 \end_layout
 
 \end_deeper
-\begin_layout Section
-Divisibilidad
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+Todo dominio con un número finito de ideales es un cuerpo, y en particular
+ lo es todo dominio finito.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
@@ -2157,6 +2362,71 @@ propio
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+Dados anillos 
+\begin_inset Formula $A_{1},\dots,A_{n}$
+\end_inset
+
+, 
+\begin_inset Formula ${\cal L}(A_{1}\times\dots\times A_{n})=\{I_{1}\times\dots\times I_{n}\}_{I_{i}\trianglelefteq A_{i},\forall i}$
+\end_inset
+
+.
+\begin_inset Quotes crd
+\end_inset
+
+ 
+\begin_inset Quotes cld
+\end_inset
+
+Si 
+\begin_inset Formula $e\in A$
+\end_inset
+
+ es idempotente, para 
+\begin_inset Formula $a\in A$
+\end_inset
+
+, 
+\begin_inset Formula $a\in(e)\iff a=ea$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $(e)$
+\end_inset
+
+ es un anillo con identidad 
+\begin_inset Formula $e$
+\end_inset
+
+.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 La intersección de toda familia de ideales de 
 \begin_inset Formula $A$
 \end_inset
@@ -2331,6 +2601,60 @@ ideal principal
 \end_inset
 
  son asociados.
+ 
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+Dado un anillo 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $b\in A$
+\end_inset
+
+ cancelable no invertible, 
+\begin_inset Formula $(b,X)$
+\end_inset
+
+ no es un ideal principal de 
+\begin_inset Formula $A[X]$
+\end_inset
+
+, y en particular 
+\begin_inset Formula $(X,Y)$
+\end_inset
+
+ no es principal de 
+\begin_inset Formula $A[X,Y]$
+\end_inset
+
+.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
@@ -2438,6 +2762,43 @@ dominio de ideales principales
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+En un DIP, 
+\begin_inset Formula $(a)+(b)=(\gcd\{a,b\})$
+\end_inset
+
+ y 
+\begin_inset Formula $(a)\cap(b)=(\text{lcm}\{a,b\})$
+\end_inset
+
+.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 No todos los ideales son finitamente generados.
  En efecto, dado un anillo no trivial 
 \begin_inset Formula $A$
@@ -2524,6 +2885,7 @@ Dados subconjuntos
 \end_inset
 
 .
+ 
 \end_layout
 
 \begin_layout Standard
@@ -2700,6 +3062,76 @@ Sea
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $I,J\trianglelefteq A$
+\end_inset
+
+ tienen 
+\series bold
+suma directa
+\series default
+ 
+\begin_inset Formula $K\trianglelefteq A$
+\end_inset
+
+, 
+\begin_inset Formula $I\oplus J=K$
+\end_inset
+
+, si 
+\begin_inset Formula $I+J=K$
+\end_inset
+
+ e 
+\begin_inset Formula $I\cap J=0$
+\end_inset
+
+.
+ 
+\begin_inset Formula $I\oplus J=A$
+\end_inset
+
+ si y sólo si existe un idempotente 
+\begin_inset Formula $e\in A$
+\end_inset
+
+ con 
+\begin_inset Formula $I=(e)$
+\end_inset
+
+ y 
+\begin_inset Formula $J=(1-e)$
+\end_inset
+
+.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 El 
 \series bold
 ideal producto
@@ -2907,6 +3339,60 @@ Los elementos de
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+Dados un DIP 
+\begin_inset Formula $A$
+\end_inset
+
+ e 
+\begin_inset Formula $I,J_{1},J_{2}\trianglelefteq A$
+\end_inset
+
+ con 
+\begin_inset Formula $I\neq0$
+\end_inset
+
+ e 
+\begin_inset Formula $IJ_{1}=IJ_{2}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $J_{1}=J_{2}$
+\end_inset
+
+.
+ Esto no es cierto en general si 
+\begin_inset Formula $A$
+\end_inset
+
+ no es un DIP.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Dados 
 \begin_inset Formula $I,J\trianglelefteq A$
 \end_inset
@@ -2984,6 +3470,62 @@ Para
 .
 \end_layout
 
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+Un anillo 
+\begin_inset Formula $A$
+\end_inset
+
+ es 
+\series bold
+completamente idempotente
+\series default
+ si todo 
+\begin_inset Formula $I\trianglelefteq A$
+\end_inset
+
+ cumple 
+\begin_inset Formula $I=I^{2}\coloneqq I\cdot I$
+\end_inset
+
+, si y sólo si para todo 
+\begin_inset Formula $I,J\trianglelefteq A$
+\end_inset
+
+ es 
+\begin_inset Formula $I\cap J=IJ$
+\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
 Isomorfía
 \end_layout
@@ -3660,6 +4202,55 @@ comaximales
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $I,J\trianglelefteq A$
+\end_inset
+
+ son comaximales si y sólo si 
+\begin_inset Formula $\forall x,y\in A,(x+I)\cap(y+J)\neq\emptyset$
+\end_inset
+
+, en cuyo caso para 
+\begin_inset Formula $n,m\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $I^{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $J^{m}$
+\end_inset
+
+ son comaximales.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 
 \series bold
 Teorema chino de los restos:
@@ -3955,7 +4546,7 @@ Veamos ahora que
 
 \end_deeper
 \begin_layout Section
-Ideales notables
+Ideales maximales
 \end_layout
 
 \begin_layout Standard
@@ -4050,6 +4641,59 @@ espectro maximal
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $I\trianglelefteq A$
+\end_inset
+
+ es maximal en 
+\begin_inset Formula $A$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $I+(X)$
+\end_inset
+
+ lo es en 
+\begin_inset Formula $A[X]$
+\end_inset
+
+, pero 
+\begin_inset Formula $I[X]$
+\end_inset
+
+ nunca es maximal en 
+\begin_inset Formula $A[X]$
+\end_inset
+
+.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Una 
 \series bold
 cadena
@@ -4192,6 +4836,213 @@ Demostración:
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+Llamamos 
+\series bold
+radical de Jacobson
+\series default
+ de un anillo 
+\begin_inset Formula $A$
+\end_inset
+
+ a 
+\begin_inset Formula $\text{Jac}(A)\coloneqq\bigcap\text{MaxSpec}(A)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall a\in A,(1+(a)\subseteq A^{*}\implies a\in\text{Jac}(A))$
+\end_inset
+
+, y en particular 
+\begin_inset Formula $\text{Nil}(A)\subseteq\text{Jac}(A)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{Jac}(A)$
+\end_inset
+
+ no contiene elementos idempotentes no nulos.
+\end_layout
+
+\begin_layout Standard
+Un anillo 
+\begin_inset Formula $A$
+\end_inset
+
+ es 
+\series bold
+local
+\series default
+ si tiene un único ideal maximal 
+\begin_inset Formula $M$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $A\setminus A^{*}$
+\end_inset
+
+ es un ideal, en cuyo caso 
+\begin_inset Formula $M=A\setminus A^{*}$
+\end_inset
+
+.
+ Entonces decimos que 
+\begin_inset Formula $(A,M)$
+\end_inset
+
+ o 
+\begin_inset Formula $(A,M,A/M)$
+\end_inset
+
+ es un 
+\series bold
+anillo local
+\series default
+.
+ Si 
+\begin_inset Formula $(A,M)$
+\end_inset
+
+ es un anillo local, 
+\begin_inset Formula $1+M$
+\end_inset
+
+ es un subgrupo multiplicativo de 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $p\in\mathbb{Z}^{+}$
+\end_inset
+
+ primo y 
+\begin_inset Formula $\mathbb{Z}_{(p)}$
+\end_inset
+
+ el subconjunto de 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ de los racionales en cuya expresión como fracción irreducible el denominador
+ no es múltiplo de 
+\begin_inset Formula $p$
+\end_inset
+
+, entonces 
+\begin_inset Formula $(\mathbb{Z}_{(p)},(\frac{p}{1}))$
+\end_inset
+
+ es un subanillo local de 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ con 
+\begin_inset Formula $\mathbb{Z}_{(p)}/(\frac{p}{1})\cong\mathbb{Z}_{p}$
+\end_inset
+
+, y es un DFU en el que 
+\begin_inset Formula $p$
+\end_inset
+
+ es el único irreducible salvo asociados.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $I\trianglelefteq A$
+\end_inset
+
+ es 
+\series bold
+nil
+\series default
+ si está contenido en 
+\begin_inset Formula $\text{Nil}(A)$
+\end_inset
+
+, y en tal caso:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall a\in A,(a+I\in(A/I)^{*}\implies a\in A^{*})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A/I$
+\end_inset
+
+ no tiene idempotentes distintos de 
+\begin_inset Formula $\overline{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $\overline{1}$
+\end_inset
+
+, tampoco los tiene 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $I$
+\end_inset
+
+ es maximal, 
+\begin_inset Formula $A$
+\end_inset
+
+ es un anillo local.
+\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
+Ideales primos
+\end_layout
+
+\begin_layout Standard
 \begin_inset Formula $I\triangleleft A$
 \end_inset
 
@@ -4568,6 +5419,133 @@ Para
 
 \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
+3.
+\end_layout
+
+\end_inset
+
+Si todo ideal principal de 
+\begin_inset Formula $A$
+\end_inset
+
+ es primo, 
+\begin_inset Formula $A$
+\end_inset
+
+ es un cuerpo.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+4.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $\forall x\in A,\exists k\geq2:x^{k}=x$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\text{Spec}(A)=\text{MaxSpec}(A)$
+\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 $I\trianglelefteq A$
+\end_inset
+
+ es primo si y sólo si lo es 
+\begin_inset Formula $I[X]$
+\end_inset
+
+ en 
+\begin_inset Formula $A[X]$
+\end_inset
+
+, si y sólo si lo es 
+\begin_inset Formula $I+(X)$
+\end_inset
+
+ en 
+\begin_inset Formula $A[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dados un homomorfismo 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ y 
+\begin_inset Formula $P\trianglelefteq_{\text{p}}B$
+\end_inset
+
+, 
+\begin_inset Formula $f^{-1}(P)\trianglelefteq_{\text{p}}A$
+\end_inset
+
+, y el recíproco se cumple si 
+\begin_inset Formula $f$
+\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
 Dado un conjunto ordenado 
 \begin_inset Formula $(S,\leq)$
 \end_inset
@@ -4783,6 +5761,10 @@ Demostración:
 .
 \end_layout
 
+\begin_layout Section
+Radicales
+\end_layout
+
 \begin_layout Standard
 \begin_inset Formula $I\trianglelefteq A$
 \end_inset
@@ -4911,13 +5893,6 @@ radical
 \end_layout
 
 \begin_layout Standard
-\begin_inset Newpage pagebreak
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
 Propiedades:
 \end_layout