diff options
Diffstat (limited to 'ac')
| -rw-r--r-- | ac/n1.lyx | 214 |
1 files changed, 150 insertions, 64 deletions
@@ -3481,32 +3481,28 @@ begin{exinfo} \end_inset -Dados un DIP +Dados un dominio \begin_inset Formula $A$ \end_inset - e -\begin_inset Formula $I,J_{1},J_{2}\trianglelefteq A$ -\end_inset - - con -\begin_inset Formula $I\neq0$ +, +\begin_inset Formula $a,b\in A$ \end_inset e -\begin_inset Formula $IJ_{1}=IJ_{2}$ +\begin_inset Formula $I\trianglelefteq A$ \end_inset -, entonces -\begin_inset Formula $J_{1}=J_{2}$ + no trivial, si +\begin_inset Formula $I(a)=I(b)$ \end_inset -. - Esto no es cierto en general si -\begin_inset Formula $A$ + entonces +\begin_inset Formula $(a)=(b)$ \end_inset - no es un DIP. +. + Esto no es cierto en general cuando los ideales no son principales. \begin_inset ERT status open @@ -4171,7 +4167,7 @@ Teoremas de isomorfía: \end_layout \begin_layout Enumerate -Para un isomorfismo de anillos +Para un homomorfismo de anillos \begin_inset Formula $f:A\to B$ \end_inset @@ -5101,28 +5097,15 @@ radical de Jacobson \end_inset a -\begin_inset Formula $\text{Jac}(A)\coloneqq\bigcap\text{MaxSpec}(A)$ +\begin_inset Formula $\text{Jac}(A)\coloneqq\bigcap\text{MaxSpec}(A)=\{a\in A:1+(a)\subseteq A^{*}\}$ \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. + no contiene idempotentes no nulos. \end_layout \begin_layout Standard @@ -5173,7 +5156,6 @@ anillo local \end_inset . - \end_layout \begin_layout Standard @@ -5251,6 +5233,142 @@ Si \end_layout \begin_layout Standard +Dados anillos locales +\begin_inset Formula $A_{1},\dots,A_{n}$ +\end_inset + +, los idempotentes de +\begin_inset Formula $A_{1}\times\dots\times A_{n}$ +\end_inset + + son las +\begin_inset Formula $n$ +\end_inset + +-uplas +\begin_inset Formula $(e_{1},\dots,e_{n})$ +\end_inset + + con cada +\begin_inset Formula $e_{i}\in\{0,1\}$ +\end_inset + +. + Para +\begin_inset Formula $n\geq2$ +\end_inset + + con factorización prima +\begin_inset Formula $p_{1}^{m_{1}}\cdots p_{t}^{m_{t}}$ +\end_inset + + (con los +\begin_inset Formula $p_{i}$ +\end_inset + + distintos y los +\begin_inset Formula $t_{i}\geq1$ +\end_inset + +), +\begin_inset Formula $\mathbb{Z}_{n}$ +\end_inset + + tiene +\begin_inset Formula $2^{t}$ +\end_inset + + idempotentes dados por los sistemas de ecuaciones diofánticas +\begin_inset Formula +\[ +\left\{ \begin{array}{rl} +e_{I} & \equiv0\mod\left(q\coloneqq\prod_{i\in I}p_{i}^{m_{i}}\right),\\ +e_{I} & \equiv1\mod\left(r\coloneqq\prod_{i\notin I}p_{i}^{m_{i}}\right), +\end{array}\right. +\] + +\end_inset + +para +\begin_inset Formula $I\subseteq\{1,\dots,t\}$ +\end_inset + +. + En concreto existen +\begin_inset Formula $s,t\in\mathbb{Z}$ +\end_inset + + con +\begin_inset Formula $x=1+qt=rs$ +\end_inset + +, de modo que +\begin_inset Formula $rs-qt=1$ +\end_inset + + y, como +\begin_inset Formula $q$ +\end_inset + + y +\begin_inset Formula $r$ +\end_inset + + son coprimos, se pueden obtener +\begin_inset Formula $s$ +\end_inset + + y +\begin_inset Formula $t$ +\end_inset + + con una identidad de Bézout. + Para obtener una identidad de Bézout: +\end_layout + +\begin_layout Enumerate +Se calcula el máximo común divisor por el algoritmo de Euclides, haciendo + +\begin_inset Formula $q_{0}\coloneqq q$ +\end_inset + +, +\begin_inset Formula $q_{1}\coloneqq r$ +\end_inset + + y la recurrencia +\begin_inset Formula $q_{i-1}=r_{i}q_{i}+q_{i+1}$ +\end_inset + +, con +\begin_inset Formula $r_{i},q_{i+1}\in\mathbb{Z}$ +\end_inset + + y +\begin_inset Formula $0\leq q_{i+1}<q_{i}$ +\end_inset + +, hasta llegar a un +\begin_inset Formula $q_{n}=1$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Se va despejando hacia atrás, haciendo +\begin_inset Formula +\begin{multline*} +1=q_{n}=q_{n-2}-r_{n-1}q_{n-1}=q_{n-2}-r_{n-1}(q_{n-3}-r_{n-2}q_{n-2})=\\ +=-r_{n-1}q_{n-3}+(1+r_{n-1}r_{n-2})q_{n-2}=\dots=q_{0}t+q_{1}s. +\end{multline*} + +\end_inset + + +\end_layout + +\begin_layout Standard \begin_inset Formula $I\trianglelefteq A$ \end_inset @@ -6247,22 +6365,6 @@ radical \end_layout \begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -begin{samepage} -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard Propiedades: \end_layout @@ -6338,22 +6440,6 @@ Sea \end_deeper \begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{samepage} -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard Un \series bold subconjunto multiplicativo |